Suppose Fred and Ethel need to divide 20 pizza slices and 10 cans of beverage. Fred's utility function is given by the equation U=Min(Z,B), where Z is the amount of pizza and B is the amount of beverage. Ethel's utility is given by the equation U=Min(1/2Z,B). What are the efficient allocation of Pizza and Beverage?
So for Fred, Z and B are efficiently allocated when Z=B. Further, there are no diminishing returns. That is MU at Z0=B0 is the same as MU at Z1=B1. For Ethel, like Fred, there are no diminishing returns. However, efficient allocation calls for 2Z=B.
So, it may not be equitable, but one efficient allocation is to give everything to Ethel.
To find the efficient allocation of pizza and beverage, we need to allocate the resources in a way that maximizes the total utility for both Fred and Ethel.
Let's start by maximizing Fred's utility function, U=Min(Z,B).
Since Fred's utility function takes the minimum value between pizza slices (Z) and cans of beverage (B), we can allocate the resources in a way that minimizes the minimum value.
In this case, since Fred's utility function is U=Min(Z,B), we should allocate the resources equally to make Z = B. So, let's divide the available resources equally among Fred and Ethel.
20 pizza slices divided equally between Fred and Ethel would be 20/2 = 10 pizza slices each.
10 cans of beverage divided equally between Fred and Ethel would be 10/2 = 5 cans of beverage each.
Now, let's calculate the utility for Fred using his utility function U=Min(Z,B).
Fred's utility would be U=Min(10, 5) = 5.
Next, let's calculate the utility for Ethel using her utility function U=Min(1/2Z,B).
Ethel's utility would be U=Min(1/2 * 10, 5) = Min(5, 5) = 5.
Therefore, the efficient allocation of pizza and beverage that maximizes the total utility for both Fred and Ethel is 10 pizza slices and 5 cans of beverage each.