Suppose that Bridget and Erin spend their incomes on two goods, food \((F)\) and clothing (C). Bridget's preferences are represented by the utility function \(U(F, C)=10 F C,\) while Erin's preferences are represented by the utility function \(U(F, C)=.20 F^{2} C^{2}\) a. With food on the horizontal axis and clothing on the vertical axis, identify on a graph the set of points that give Bridget the same level of utility as the bundle \((10,5) .\) Do the same for Erin on a separate graph. b. On the same two graphs, identify the set of bundles that give Bridget and Erin the same level of utility as the bundle (15,8) c. Do you think Bridget and Erin have the same preferences or different preferences? Explain.

Step by step solution

01

Identifying the Utility Bundles

Firstly, we substitute the provided bundle into each of their utility functions. For Bridget, using the bundle (10,5), this gives: \(U(F, C) = 10 * 10 * 5 = 500\). For Erin, using the same bundle, this gives: \(U(F, C) = 0.2 * 10^{2} * 5^{2} = 200\). We will use these utility values to find the points that provide the same level of utility on the graph for individual.

02

Graphing the Utility Levels

Each point on the graph that gives the same utility corresponds to the same bundle. Thus, for Bridget, the graph will be a curve coming from the equation derived from her utility function: \(10FC = 500\). This would be a hyperbola. Similarly, for Erin, it will be a curve from: \(0.2F^{2}C^{2} = 200\). This would be a parabola. Each bundle on this curve gives the individuals the same level of utility.

03

Identifying Bundles with Utility (15,8)

We now repeat the steps for the bundle (15,8). For Bridget, \(U(F, C) = 10*15*8 = 1200\). For Erin, \(U(F, C) = 0.2 * 15^{2} * 8^{2} = 720\), and we can represent these on the same individual graphs.

04

Analyzing Preferences

Bridget's and Erin's curves give us the same utility levels but are different shapes due to the different utility functions. They do not have the same preferences, as indicated by their distinct utility functions. Each bundle would give them different utility levels.

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