Please Complete The Following Two Applied Problems Problem 1

Please Complete The Followingtwo Applied Problemsproblem 1william I

Please complete the following two applied problems: Problem 1: William is the owner of a small pizza shop and is thinking of increasing products and lowering costs. William’s pizza shop owns four ovens and the cost of the four ovens is $1,000. Each worker is paid $500 per week. Show all of your calculations and processes. Describe your answer for each question in complete sentences, whenever it is necessary.

Which inputs are fixed and which are variable in the production function of William’s pizza shop? Over what ranges do there appear to be increasing, constant, and/or diminishing returns to the number of workers employed? What number of workers appears to be most efficient in terms of pizza product per worker? What number of workers appears to minimize the marginal cost of pizza production assuming that each pizza worker is paid $500 per week? Why would marginal productivity decline when you hire more workers in the short run after a certain level?

How would expanding the business affect the economies of scale? When would you have constant returns to scale or diseconomies of scale? Describe your answer. Problem 2: The Paradise Shoes Company has estimated its weekly TVC function from data collected over the past several months, as TVC = 3450 + 20Q + 0.008Q² where TVC represents the total variable cost and Q represents pairs of shoes produced per week. And its demand equation is Q = 4100 – 25P.

The company is currently producing 1,000 pairs of shoes weekly and is considering expanding its output to 1,200 pairs of shoes weekly. To do this, it will have to lease another shoe-making machine ($2,000 per week fixed payment until the lease period ends). Show all of your calculations and processes. Describe and derive an expression for the marginal cost (MC) curve.

Describe and estimate the incremental costs of the extra 200 pairs per week (from 1,000 pairs to 1,200 pairs of shoes). What are the profit-maximizing price and output levels for Paradise Shoes? Describe and calculate the profit-maximizing price and output. Discuss whether or not Paradise Shoes should expand its output further beyond 1,200 pairs per week. State all assumptions and qualifications that underlie your recommendation.

Paper For Above instruction

The provided problems explore fundamental concepts of microeconomics including production functions, returns to scale, cost analysis, and profit maximization, all within practical business contexts. In Problem 1, the focus is on a small pizza shop owned by William, examining the relationship between inputs such as labor and capital (ovens), and the resulting output and costs. Understanding fixed versus variable inputs is essential to analyzing operational efficiency, marginal productivity, and economies of scale. Problem 2 centers on the shoe manufacturing business, analyzing its cost structure and demand to determine profit-maximizing output and prices, as well as the implications of expanding production.

Problem 1: William’s Pizza Shop – Inputs, Production, and Economies of Scale

William’s pizza shop operates with four ovens and employs labor—a typical scenario reflecting fixed and variable inputs. The four ovens, costing a total of $1,000, are fixed inputs because their number and cost do not change with short-term variations in output. The labor, paid weekly at $500 per worker, is a variable input because the number of workers can be adjusted based on production needs.

In analyzing the production process, the total output (number of pizzas) depends on the combination of fixed and variable inputs. The law of diminishing returns suggests that initially, adding workers will increase output at an increasing rate (increasing returns), then at a constant rate (constant returns), and eventually at a decreasing rate (diminishing returns). Empirical data or a firm’s production function would enable precise identification of these ranges. Typically, the most efficient number of workers is where marginal product per worker is maximized, which occurs before diminishing returns set in.

To determine the optimal employment level, the firm must compare the marginal product of labor (MPL) with the marginal cost (MC) of labor. The MC of each worker is $500 per week, and the most efficient level of employment is found where the MPL per worker yields the lowest average cost per pizza. As more workers are hired, the MPL tends to decline after a certain point — due to limited capital or operational constraints — which results in the law of diminishing marginal returns.

Expanding the business impacts economies of scale. In the short run, increasing output may initially benefit from economies of scale—cost advantages as production expands—mainly through spreading fixed costs like ovens over more output. However, once the firm hits limits of capacity or managerial efficiency diminishes, it experiences diseconomies of scale, where costs per unit rise. Constant returns to scale occur when increasing input proportionally results in a proportional increase in output without cost advantages or disadvantages.

Problem 2: Cost Structure and Profit Maximization for Paradise Shoes

The weekly total variable cost function of Paradise Shoes is given by TVC = 3450 + 20Q + 0.008Q², where Q is the number of shoe pairs produced weekly. The demand equation Q = 4100 – 25P ties quantity to price. Currently, the firm produces 1,000 pairs of shoes weekly and considers increasing output to 1,200 pairs, which entails leasing an additional machine costing $2,000 weekly.

An essential step is deriving the marginal cost (MC) curve, which indicates the cost of producing each additional unit of output. The MC is the first derivative of the total cost (TC), which includes variable and fixed costs. The total cost function can be written as TC = TVC + TFC; since TFC is fixed at $3,450 plus the additional lease cost when applicable, the MC for the additional units can be derived from the variable cost component.

Calculating the incremental cost of expanding from 1,000 to 1,200 pairs involves finding the difference in total variable costs at these quantities. The marginal cost at each level of output informs optimal pricing and quantity decisions. The profit-maximizing output occurs where marginal revenue (derived from the demand curve) equals marginal cost. Substituting the demand equation into revenue calculations allows solving for the optimal price at that quantity.

If expanding beyond 1,200 pairs, the firm should consider whether the additional costs (including leasing the new machine) are justified by the increased revenue. If the incremental revenue exceeds incremental costs, further expansion may be profitable; otherwise, it might not be viable. Assumptions underlying these analyses include perfect competition, stable demand, and fixed costs remaining constant aside from the new lease.

Conclusion

Both problems illustrate critical economic principles in a practical setting: how fixed and variable inputs determine production efficiency, how costs influence decision-making, and how firms maximize profits in competitive markets. Carefully analyzing these factors aids managers in making informed choices about optimal input levels, scaling strategies, and pricing policies.

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