Which inputs are fixed and which are variable in the production function of William’s pizza shop?

  

Please complete the followingtwo applied problems:Problem 1:William is the owner of a small pizza shop and is thinking of increasing products and lowering costs. Williams pizza shop owns four ovens and the cost of the four ovens is $1,000. Each worker is paid $500 per week.Workers employedQty of pizzas produced per week012345678075180360600900114012601360Show 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 Williams 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 your answer for each item below in complete sentences, whenever it is necessary.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.

Introduction:

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In these applied problems, we will be discussing the production function of William’s pizza shop and the cost and profit analysis of the Paradise Shoes Company. In problem 1, we will examine the relationship between the inputs and outputs of William’s pizza shop, including the fixed and variable costs, the marginal cost of pizza production, and the reasons behind the decline of marginal productivity. In problem 2, we will analyze the marginal cost curve of the Paradise Shoes Company, estimate the incremental costs of expanding the output, and determine the profit-maximizing price and output levels.

Problem 1:

Inputs and Outputs of William’s Pizza Shop:

William is the owner of a small pizza shop and is considering expanding his production while reducing costs. He has four ovens, which cost him $1,000, while employee wages are $500 per week. To determine the relationship between the inputs and outputs, we will examine the quantity of pizzas produced per week based on the number of workers employed. The results are as follows:

| Workers Employed | Qty of Pizzas Produced per Week |
|—————–|——————————–|
| 0 | 0 |
| 1 | 75 |
| 2 | 180 |
| 3 | 360 |
| 4 | 600 |
| 5 | 900 |
| 6 | 1140 |
| 7 | 1260 |
| 8 | 1360 |

Fixed and Variable Costs:

William’s pizza shop has two types of costs: fixed cost and variable cost. Fixed costs include the cost of four ovens, which is $1,000, and cannot be changed in the short run. On the other hand, variable costs include the wages paid to workers, which is $500 per week, and are directly proportional to the quantity of pizzas produced. Hence, fixed cost is independent of output, while variable cost is dependent on output.

Marginal Cost of Pizza Production:

The marginal cost of pizza production is the additional cost incurred to produce one more pizza. The formula for marginal cost is the change in total cost divided by the change in quantity. The results for William’s pizza shop are as follows:

| Workers Employed | Qty of Pizzas Produced per Week | Total Cost (TC) | Marginal Cost (MC) |
|—————–|——————————–|—————-|——————-|
| 0 | 0 | 1000 | – |
| 1 | 75 | 1500 | 20 |
| 2 | 180 | 2000 | 8.33 |
| 3 | 360 | 2500 | 5.56 |
| 4 | 600 | 3000 | 8.33 |
| 5 | 900 | 3750 | 12.5 |
| 6 | 1140 | 4500 | 10 |
| 7 | 1260 | 5250 | 8.33 |
| 8 | 1360 | 6000 | 12.5 |

From the above table, it can be observed that marginal cost initially decreases and then increases as the number of workers is increased, indicating that after a certain point, hiring additional workers results in diminishing returns to labor.

Economies of Scale:

Expanding the business of William’s pizza shop could result in economies of scale, which occur when the cost per unit of production decreases as the scale of production increases. However, if the cost per unit of production increases with a rise in output, it is termed diseconomies of scale. In the case of William’s pizza shop, the economies of scale would depend on the nature of the inputs, technology, and managerial efficiency.

Problem 2:

Marginal Cost and Incremental Costs:

The Paradise Shoes Company has estimated its weekly total variable cost (TVC) function and demand equation, which is given as TVC = 3450 + 20Q + 0.008Q^2 and Q = 4100 – 25P, respectively. The company is currently producing 1,000 pairs of shoes weekly and is considering increasing its output to 1,200 pairs of shoes weekly. To do so, it will have to lease another shoe-making machine at a fixed cost of $2,000 per week.

Marginal Cost:

The marginal cost (MC) curve represents the additional cost required to produce one more unit of output. For the Paradise Shoes Company, the marginal cost can be calculated by taking the derivative of the TVC function with respect to Q, i.e., MC = dTVC/dQ = 20 + 0.016Q. The marginal cost curve is upward-sloping because of increasing marginal returns to labor and constant marginal returns to the variable inputs.

Incremental Costs and Profit-Maximizing Price and Output:

To determine the incremental costs of producing an additional 200 pairs of shoes, we can use the marginal cost formula, which is MC = ΔTVC/ΔQ. Therefore, the incremental cost would be MC × ΔQ = (20 + 0.016Q) × 200 = $4,016. The profit-maximizing price and output can be determined by setting MR = MC, where MR is the marginal revenue, and solving the demand equation for P. The profit-maximizing output is 2,000 pairs of shoes, and the profit-maximizing price is $80 per pair. The total profit would be $40,000.

Expanding Output:

Whether or not the Paradise Shoes Company should expand its output further beyond 1,200 pairs per week depends on various factors, such as the demand for shoes, availability of resources, and competition. However, it is essential to consider the diminishing returns to labor, which will result in an increase in marginal cost, and the potential diseconomies of scale, which will increase the cost per unit of production with a rise in output. Additionally, the fixed cost of leasing another shoe-making machine would add to the total cost of production. Hence, the company should analyze the costs and benefits of expansion before making any decisions.

Objectives:
– To understand the concept of fixed and variable inputs in the production function
– To analyze the relationship between the number of workers employed and pizza production
– To determine the most efficient number of workers and the level at which marginal cost is minimized
– To explain the concept of diminishing marginal productivity
– To evaluate the economies of scale in expanding a business
– To understand and apply the TVC function, marginal cost curve, and profit maximization in decision making

Learning Outcomes:
By the end of this exercise, learners will be able to:
– Differentiate between fixed and variable inputs in a production function
– Analyze the relationship between inputs and outputs, and identify the ranges of increasing, constant, or diminishing returns
– Calculate the most efficient number of workers and the level at which marginal cost is minimized
– Explain the concept of diminishing marginal productivity and its implications in decision making
– Evaluate the economies of scale related to business expansion and identify the conditions under which constant returns to scale or diseconomies of scale occur
– Apply the TVC function, marginal cost curve, and profit maximization concepts to make informed decisions related to business operations

Problem 1:
Fixed inputs: Four ovens
Variable inputs: Number of workers employed
Increasing returns: 0-3 workers
Constant returns: 4-5 workers
Diminishing returns: 6-8 workers

Efficient number of workers: 6
Minimum marginal cost: 5 workers

Marginal productivity declines when you hire more workers because of the limited amount of fixed inputs (ovens), which results in a reduction in the output per worker.

Expanding the business may result in economies of scale, which lower the average cost of production. However, there could be diseconomies of scale if the cost of managing a larger operation outweighs the benefits of increased efficiency. The point at which constant returns to scale or diseconomies of scale occur varies depending on the nature of the business.

Problem 2:
MC = dTVC/dQ = 20 + 0.016Q
Incremental cost of extra 200 pairs per week = MC(1,200) – MC(1,000) = $52

Profit-maximizing output: Q = 5,500 pairs of shoes per week
Profit-maximizing price: P = $78.50 per pair

Paradise Shoes should consider expanding output beyond 1,200 pairs per week if the marginal revenue exceeds the marginal cost. However, they should also consider market demand and competition, as well as the availability of resources and capacity constraints.

Solution 1:

William is the owner of a small pizza shop and wants to increase his products while minimizing the costs. Four ovens are already owned and the cost of these ovens is $1,000. Each worker is paid $500 per week. The following table shows the workers employed and the quantity of pizzas produced per week.

Workers employed | Qty of pizzas produced per week
—|—
0 | 75
1 | 180
2 | 360
3 | 600
4 | 900
5 | 1140
6 | 1260
7 | 1360

Fixed inputs in William’s pizza shop are four ovens as they cannot be increased or decreased in the short run. Variable inputs are workers employed as they can be varied.

The production function at small pizza shops has three stages: increasing returns, diminishing returns, and negative returns. As the number of workers increases from 0 to 2, the production output increases from 75 to 360, indicating increasing returns. From 2 to 6 workers, the production output increases from 360 to 1260, but at a decreasing rate, indicating diminishing returns. Beyond six workers, the production output decreases with the addition of more workers, indicating negative returns.

The number of workers that appears to be most efficient in terms of pizza product per worker is six workers. This is because beyond six workers, the marginal product of labor decreases and as such, the average product of labor also decreases.

The marginal cost of pizza production can be computed by dividing the change in total cost by the change in the quantity produced. Each pizza worker is paid $500 per week, and adding one worker increases the total cost by this amount. The marginal product of labor (MPL) can also be computed by dividing the change in the quantity produced by the change in the number of workers. The marginal cost (MC) is then the change in total cost divided by the change in quantity.

At the sixth worker, the marginal cost of pizza production is at a minimum. Beyond six workers, the marginal cost of pizza production starts to increase.

In the short run, marginal productivity will eventually decline when you hire more workers after a certain level because when there are too many workers assigned to a fixed amount of capital, then the productivity of each additional worker will fall.

Expanding the business can affect economies of scale positively if the increased level of output leads to a decrease in average cost. Constant returns to scale will be observed when output increases proportionately to the increase in all inputs. Diseconomies of scale occur when a firm experiences a rise in the average cost when output is increased beyond a certain point.

Solution 2:

The Paradise Shoes Company has estimated its weekly total variable cost (TVC) function from data collected over the past several months, as TVC = 3450 + 20Q + 0.008Q^2. The demand equation for Paradise Shoes 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 by leasing another shoe-making machine, costing $2,000 per week fixed payment until the lease period ends.

To derive the marginal cost (MC) curve, we need to calculate the first derivative of the TVC function.

TVC = 3450 + 20Q + 0.008Q^2
MC = dTVC/dQ = 20 + 0.016Q

The incremental costs of producing the extra 200 pairs of shoes per week can be estimated by finding the TVC for that level of output.

TVC = 3450 + 20(1200) + 0.008(1200)^2 – (3450 + 20(1000) + 0.008(1000)^2)
TVC = $2,560

To determine the profit-maximizing price and output levels for Paradise Shoes, we need to find the level of output at which marginal cost equals marginal revenue (MR).

MR = dQ/dP * (P/Q)
MR = (4100 – 50P) / 25

Setting MC equal to MR, we get:

20 + 0.016Q = (4100 – 50P) / 25

Solving for Q and P, we get:

Q = 6625
P = $102.50

Therefore, the profit-maximizing price and output levels for Paradise Shoes are 6,625 pairs of shoes at a price of $102.50 per pair.

Whether or not Paradise Shoes should expand its output further beyond 1,200 pairs per week depends on the market demand. If there is a market demand for additional shoes, then expanding output further could lead to increased profits. However, if demand is not there, then expanding output could lead to excess inventory and decreased profitability. The decision to expand should be based on a thorough analysis of market demand and the cost-benefit trade-offs of different output levels.

Suggested Resources/Books:
1. Managerial Economics and Business Strategy by Michael Baye
2. Microeconomics by Robert Pindyck and Daniel Rubinfeld
3. Production and Operations Management by Martin K. Starr

Similar Asked Questions:
1. What are some strategies for increasing product and lowering costs in a small business?
2. How does productivity change with an increase in the number of workers in the production process?
3. What are the factors affecting economies of scale in a business?
4. How do fixed and variable costs impact a company’s profitability?
5. What is the relationship between the marginal cost and total variable cost in a production function?

Problem 1:
Inputs: The cost of the four ovens is fixed, and the number of workers employed is variable.
– Increasing Returns: There are increasing returns up to six workers, where pizzas produced per worker increases.
– Constant Returns: There are constant returns between six and eight workers, where each additional worker produces approximately the same number of pizzas.
– Diminishing Returns: There are diminishing returns from eight to twelve workers, where pizzas produced per worker decrease.

The number of workers that appears to be most efficient is six workers since they produce 180 pizzas per worker.

The number of workers that appears to minimize the marginal cost of pizza production is eight workers as the additional cost of hiring an extra worker is higher than the additional pizzas produced beyond the eighth worker.

Marginal productivity declines when you hire more workers in the short-run since there is limited space and machinery, which leads to congestion and increased waiting time, resulting in a decrease in productivity.

Expanding the business can affect economies of scale either positively, leading to lower costs of production or negatively, leading to higher costs of production. Constant returns to scale occur when the cost of production remains the same after an increase in output. Diseconomies of scale occur when the cost of production increases after an increase in output due to inefficiencies and management difficulties.

Problem 2:
The marginal cost curve (MC) can be derived by calculating the first derivative of the TVC function, which is MC = 20 + 0.016Q.

The incremental cost of producing an additional 200 pairs per week is the marginal cost multiplied by the increase in output, which is (20 + 0.016(1,200)) – (20 + 0.016(1,000)) = $64.

The profit-maximizing price and output levels can be calculated by finding the level of output where marginal cost equals marginal revenue (MR), and then finding the corresponding price using the demand function. The profit-maximizing output is 3,750 pairs of shoes per week, and the corresponding price is $95. The total profit at this level is $123,125.

Paradise Shoes should consider expanding its output beyond 1,200 pairs per week, given that the marginal revenue from each additional shoe produced is greater than its incremental cost, resulting in a higher profit margin. However, this assumption depends on the demand remaining constant and no other costs incurred from expanding the operation. Therefore, a thorough cost-benefit analysis is recommended to assess the feasibility of expanding further.

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