Public Good. Suppose there are two goods X and Y in the economy where X is a public good and Y is a private good. Let X + Y = 100 denote the Production Possibility Frontier (PPF). There are two individuals A and B with utility functions uA(X, YA) = XYA and uB(X, YB) = XYB. Note that X does not have a subscript because it is a public good and the two individuals have to consume the same amount. Y is a private good so Y = YA + YB. (a) Suppose A is to achieve a utility of ¯uA. Find the amount of YB available to individual B as a function of different values of X. (b) Find the amount of X and YB that will maximize individual B’s utility. This could depend on ¯uA. Find individual A’s consumption of YA as well. (c) What is the condition for Pareto efficient allocation in such a case? Is the answer you find in part (b) Pareto efficient.

Part a: The value of Y_A and Y_B are and respectively.

Part b: Y_A and Y_B are given as and respectively for maximization of Y_B

Part c: The condition for the Pareto efficient allocation is Y_A=Y_B

As the value of Y_A and Y_B are not equal in part 2 thus the condition is not Pareto efficient

Explanation:

Part a

For the value of the utility function is given as

Also the YB is given as

So the value of Y_A and Y_B are and respectively.

Part b:

Now

For the maximization

From question 1 Y_A and Y_B are given as and respectively for maximization of Y_B

Part c:

At the Pareto efficient allocation

This is simplified to

The condition for the Pareto efficient allocation is YA=YB

As the value of YA and YB are not equal in part 2 thus the condition is not Pareto efficient

1. avoid them

2. a talk with them

3. ask someone you trust

4. ignore

5. call police

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