Consider a coffee shop allocating 30 workers between two shops. The marginal product of workers at shop one is MPL-100-2L, and the marginal product of workers at shop two is MPL2-70-L2. What is the optimal amount of workers to send
to the second shop?
15
20
5
10
i am requesting please do not use chatgpt and do not give handwritten solution
In the dynamic and competitive world of coffee shop management, efficient allocation of human resources is paramount to success. Deciding how many workers to allocate to each shop can significantly impact productivity and ultimately, profitability. This essay explores the optimization of worker allocation for two coffee shops, leveraging the concept of marginal product of labor (MPL) to make informed decisions.
Before delving into the optimization process, it is crucial to comprehend the concept of MPL. MPL represents the additional output achieved by adding one more worker to the workforce. In our scenario, Shop One’s MPL is expressed as MPL1 = 100 – 2L, where ‘L’ denotes the number of workers allocated to the first shop. Similarly, Shop Two’s MPL is described as MPL2 = 70 – L^2, where ‘L’ signifies the number of workers sent to the second shop.
To allocate workers optimally, our goal is to maximize total output, which is the combined productivity of both coffee shops. The output of each shop, Q1 and Q2, can be calculated using the respective MPL functions:
Q1 = MPL1 * L Q2 = MPL2 * (30 – L) (as there are a total of 30 workers available)
To find the optimal allocation, we must identify the value of ‘L’ that maximizes the total output, Q. This involves taking the derivative of the total output with respect to ‘L’ and setting it equal to zero, resulting in a critical point.
Our optimization process begins by differentiating both Q1 and Q2 with respect to ‘L’:
dQ1/dL = (100 – 2L) dQ2/dL = (70 – L^2)
Setting these derivatives equal to zero, we can solve for ‘L.’ The critical point for Shop One (dQ1/dL = 0) yields L = 50, while for Shop Two (dQ2/dL = 0), we get L = ±√70. Since we cannot allocate a negative number of workers, we discard the negative solution.
Therefore, the optimal number of workers to allocate to the second shop is approximately 8 (rounded to the nearest whole number). This allocation ensures the highest overall productivity, taking into account the unique MPL characteristics of both shops.
Efficiently allocating workers between two coffee shops is a vital decision for coffee shop owners seeking to maximize productivity and profitability. Leveraging the concept of MPL and optimizing worker allocation based on marginal product of labor functions can lead to informed decisions. In our scenario, the optimal allocation of approximately 8 workers to the second shop promises to yield the best overall outcomes, ensuring that coffee shops can meet customer demands and excel in the competitive market. By embracing data-driven approaches like this, coffee shop owners can take confident strides towards business success in a rapidly evolving industry.
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