# Utility, Preference maps, Edgeworth Box, and Contract curve (Microeconomics question)

Imagine the following world: There are two members of society, Wayne and Garth. There are two goods in the world: cups of Coffee (C) and Donuts (D). There are 100 cups of Coffee and 100 Donuts. Wayne likes to drink exactly 3 cups of Coffee with each Donut that he eats. Garth, however, likes to eat exactly 3 Donuts with every cup of Coffee that he drinks. Neither Wayne nor Garth get any utility from Coffee or Donuts outside their preferred ratio. Wayne’s preferences can be represented by UW (C, D) = min{C, 3D}, and Garth’s preferences can be represented by UG(C, D) = min{3C, D}. [Note that, throughout this question, cups of Coffee and Donuts are divisible. For example, it’s possible to drink 1/4 cups of Coffee and 1/2 a Donut.) Using this information, answer the following questions.

Part (a) On separate graphs, illustrate Wayne’s and Garth’s preference map over Coffee and Donuts putting Coffee on the x-axis and Donuts on the y-axis. Note: When referencing these graphs below, I’ll refer to Wayne’s preference map as A1 and Garth’s preference map as A2.

Part (b) On a different graph, illustrate the set of utility possibilities for Wayne and Garth when all Coffee and Donuts are distributed, putting Wayne’s utility on the y-axis and Garth’s utility on the x-axis. Note: When referencing this graph below, I’ll refer to it as B.

Part (c) Suppose Wayne is given all of the Donuts and Garth is given all of the Coffee. On graph B, illustrate the utility pair associated with this initial allocation. Label this point X. Is this allocation Pareto optimal? Is it efficient? Why or why not?

Part (d) On graphs A1 and A2, illustrate the initial allocation (label it X) and indicate the set of allocations that are preferred by each to this initial allocation.

Part (e) Combine graphs A1 and A2 to create an Edgeworth Box of Wayne’s and Garth’s preferences and possible trades. Put Wayne’s origin (0 Coffee and 0 Donuts) at the bottom left corner and put Garth’s origin (0 Coffee and 0 Donuts) at the top right corner. Label the initial allocation X and indi- cate the set of possible mutually beneficial trades.

Part (f) On graph B, identify and label the utility pairs associated with allocations that are Pareto preferred to X.

Part (g) On graph B, identify the set of allocations that are Pareto optimal. Label this set “Pareto Frontier”.

Part (h) On the Edgeworth box you created in Part (e), identify and label the Contract Curve. Given the initial allocation, which portion of the Contract Curve will be utilized by Wayne and Garth if they are permitted to trade?

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