Suppose 15 peaches and 20 oranges need to be divided between fred and ethel. Fred's utility function is U=Max(#of peaches, #of oranges) ethels utility function is U=Max(2# peaches, # of oranges). Carefully draw a graph showing the set of Pareto efficient allocations.
I don't understand your Utility functions. If I read your post, literally, then:
If Fred has 1 peaches and 10 oranges, his utility is same as if he had 9 peaches and 10 oranges. The utility for both mixes is 10? This does not make sense.
To draw a graph showing the set of Pareto efficient allocations, follow these steps:
Step 1: Define the axes: Label the x-axis as the number of peaches allocated to Fred, and the y-axis as the number of oranges allocated to Fred.
Step 2: Plot the utility functions:
- For Fred's utility function (U = max(# of peaches, # of oranges)):
- Draw a line from the origin (0, 0) with a slope of 1 until it intersects the x-axis (representing when Fred gets only peaches) and the y-axis (representing when Fred gets only oranges).
- For Ethel's utility function (U = max(2 * # of peaches, # of oranges)):
- Draw a line from the origin (0, 0) with a slope of 2 until it intersects the x-axis (representing when Ethel gets only peaches) and the y-axis (representing when Ethel gets only oranges).
Step 3: Identify the Pareto efficient allocations:
- Pareto efficient allocations occur where neither Fred nor Ethel can be made better off without making at least one of them worse off.
- Look for points where either Fred's utility function equals Ethel's utility function or where one utility function dominates the other.
- These points represent the set of Pareto efficient allocations.
Step 4: Mark the Pareto efficient allocations on the graph:
- Identify all the points where either Fred's utility function equals Ethel's utility function or where one utility function dominates the other.
- Mark these points on the graph with a dot or a label.
Step 5: Connect the Pareto efficient allocations:
- Once you have marked all the Pareto efficient allocations, connect them with a line or curve to show the set of Pareto efficient allocations.
By following these steps, you can carefully draw a graph displaying the set of Pareto efficient allocations for the given scenario.