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 b: Y_A and Y_B are given as [tex]\dfrac{\bar{u}_A}{50}[/tex] and [tex]50-\dfrac{\bar{u}_A}{50}[/tex] 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

[tex]\bar{u}_A=xY_A\\Y_A=\dfrac{\bar{u}_A}{x}[/tex]

Also the YB is given as

[tex]Y_A+Y_B=100-x\\Y_B=100-x-Y_A\\Y_B=100-x-\dfrac{\bar{u}_A}{x}[/tex]

So the value of Y_A and Y_B are [tex]\dfrac{\bar{u}_A}{x}[/tex] and [tex]100-x-\dfrac{\bar{u}_A}{x}[/tex] respectively.

Part b:

Now

[tex]\bar{u}_B=xY_B\\\bar{u}_B=x(100-x-\dfrac{\bar{u}_A}{x})\\\bar{u}_B=100x-x^2-\bar{u}_A[/tex]

For the maximization

[tex]\dfrac{\partial \bar{u}_B}{\partial x}=0\\\dfrac{\partial (100x-x^2-\bar{u}_A)}{\partial x}=0\\100-2x=0\\x=100/2\\x=50[/tex]

From question 1 Y_A and Y_B are given as [tex]\dfrac{\bar{u}_A}{50}[/tex] and [tex]50-\dfrac{\bar{u}_A}{50}[/tex] respectively for maximization of Y_B

Part c:

At the Pareto efficient allocation

[tex]\dfrac{Mu_X}{Mu_{Y_A}}=\dfrac{Mu_X}{Mu_{Y_B}}[/tex]

This is simplified to

[tex]\dfrac{Y_A}{x}=\dfrac{Y_B}{x}\\Y_A=Y_B[/tex]

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