Suppose that Bridget and Erin spend their incomes on two goods, food(F) and clothing (C). Bridgett’s preferences are represented by the utility function U (F,C) = 10FC, while Erin’s preferences are represented by the utility function U(F,C) = 0.20F2C2. (That’s F squared and C squared.)
a. Assume that Bridget has $1,200 and the price of food is $20, and the price of clothing is $50. Calculate the quantity of food and clothing that will maximize Bridget’s satisfaction.
b. Suppose that Erin has $2,000, the price of food is $45 and the price of clothing is $80. Calculate the quantity of food and clothing that will maximize Erin’s satisfaction.
In the world of microeconomics, consumers face choices every day, balancing their desires with budget constraints. This essay delves into the fascinating world of consumer preferences by analyzing the cases of Bridget and Erin, who have unique utility functions, budgets, and price constraints.
Bridget, with $1,200 at her disposal, aims to optimize her satisfaction while facing prices of $20 for food and $50 for clothing. Her utility function is represented as U(F, C) = 10FC. To find the combination of food and clothing that maximizes her utility, we begin by formulating her budget constraint:
20F + 50C = 1,200
Solving for F in terms of C, we find F = (1,200 – 50C)/20. Substituting this into her utility function yields U(C) = 600C – 25C^2. To maximize satisfaction, we differentiate U(C) with respect to C and set it to zero:
dU/dC = 600 – 50C = 0
Solving for C, we find C = 12. With the optimal quantity of clothing determined, we use the budget constraint to find the corresponding food quantity: F = 30. Bridget should buy 30 units of food and 12 units of clothing to maximize her satisfaction.
Erin faces a more intricate optimization problem with a $2,000 budget, $45 food price, and $80 clothing price. Her utility function is U(F, C) = 0.20F^2C^2. The budget constraint in this case is expressed as:
45F + 80C = 2,000
Solving for F in terms of C, we find F = (2,000 – 80C)/45. Substituting this into her utility function gives us a more complex U(C) equation, which we need to differentiate and set to zero to find the optimal quantity of clothing. This equation involves squared terms and some complexity, but with the right tools or software, Erin can determine the ideal quantity of clothing.
After solving for C, Erin can use her budget constraint to calculate the corresponding quantity of food, similarly to Bridget’s case.
In the realm of microeconomics, individuals like Bridget and Erin continually grapple with the choices of allocating their resources between various goods. Their unique utility functions, budgets, and price constraints add a layer of complexity to the decision-making process.
By optimizing their satisfaction, Bridget and Erin aim to balance their desires for food and clothing within their respective budgets and the prices of the goods. The process involves mathematical calculations and the application of budget constraints to find the ideal combination of goods. Bridget’s case demonstrates a simpler, linear optimization, while Erin’s presents a more complex, quadratic problem.
Ultimately, these scenarios exemplify the intriguing world of consumer choices, where mathematical tools and economic principles guide individuals in making the most of their resources and maximizing their satisfaction.
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