Question

Eleanor and her little sister Joanna are responsible for two chores on their family's farm, gathering eggs and collecting milk. Eleanor can gather 9 dozen eggs or collect 3 gallons of milk per week. Joanna can gather 2 dozen eggs or collect 2 gallons of milk per week. [LO 2.3] a. The family wants 2 gallons of milk per week and as many eggs as the sisters can gather. Currently, Eleanor and Joanna collect one gallon of milk each and as many eggs as they can. How many dozens of eggs does the family have per week? b. If the sisters specialized, which sister should gather the milk? c. If the sisters specialized, how many dozens of eggs would the family have per week?

Short answer

a) 7 dozens b) Joanna c) 9 dozens

Step-by-step solution

Understand the Resources and Needs

Eleanor can either gather 9 dozen eggs or collect 3 gallons of milk per week, while Joanna can gather 2 dozen eggs or collect 2 gallons of milk per week. The family needs 2 gallons of milk weekly.

Calculate Current Egg Production

Currently, both sisters each collect 1 gallon of milk, fulfilling the family's milk requirement of 2 gallons per week without specialization.- Eleanor is spending about 1/3 of her time collecting milk, so she gathers: \( \frac{2}{3} \times 9 = 6 \) dozen eggs per week.- Joanna is spending 1/2 of her time on milk, so she gathers: \( \frac{1}{2} \times 2 = 1 \) dozen eggs per week.- Total eggs = 6 + 1 = 7 dozen eggs per week.

Determine Specialization Based on Opportunity Cost

Calculate the opportunity cost of each doing milk:- Eleanor: 1 gallon of milk = \( \frac{9}{3} = 3 \) dozen eggs per gallon.- Joanna: 1 gallon of milk = \( \frac{2}{2} = 1 \) dozen eggs per gallon.Since Eleanor has a higher opportunity cost for milk, Joanna should specialize in milk collection.

Calculate Maximum Egg Production with Specialization

If Joanna collects all 2 gallons of milk, Eleanor can dedicate her entire time to gathering eggs: - Eleanor gathers 9 dozen eggs per week (100% egg production). - Joanna gathers 0 dozen eggs but ensures milk is collected. - Total eggs = 9 dozen eggs per week.

Key concepts

Specialization

In simple terms, specialization refers to focusing on a specific task to increase efficiency. When Eleanor and Joanna share the workload without specializing, both perform tasks less efficiently. • **Specialization Benefits:** By concentrating on what they do best, the sisters can accomplish more. Joanna can focus entirely on collecting milk since it's the less profitable task for Eleanor in terms of lost egg production. • **Improved Efficiency:** With specialization, Eleanor can devote all her efforts to egg gathering, which is her more productive task. Thus, specialization enables Eleanor and Joanna to utilize their strengths optimally and, in this case, maximize the family’s egg production.

Comparative Advantage

Comparative advantage is understanding who can perform a task at the lower opportunity cost. When the sisters consider whether Eleanor or Joanna should specialize in milk collection, it's important to look at opportunity costs. • **Eleanor's Opportunity Cost:** For every gallon of milk she collects, Eleanor forgoes 3 dozen eggs. • **Joanna's Opportunity Cost:** Meanwhile, Joanna sacrifices only 1 dozen eggs per gallon of milk she collects. Since Joanna sacrifices fewer eggs, she has a comparative advantage in milk collection. Thus, she should specialize in that task, allowing Eleanor to focus entirely on her strength, which is gathering eggs.

Resource Allocation

Resource allocation is about distributing skills and efforts to maximize outcomes. In this scenario, figuring out who should gather eggs or milk means looking at who can generate the greatest benefit from their efforts. • **Maximizing Output:** Initially, Eleanor and Joanna split the chores, leading to only 7 dozen eggs collected along with 2 gallons of milk. • **Revised Allocation:** When Joanna handles milk collection, Eleanor can gather her full potential of 9 dozen eggs, increasing total egg production. Effective resource allocation means adjusting roles so that each person plays to their strengths, which in this situation results in an increase from 7 dozen to 9 dozen eggs without decreasing milk output.