Question

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.

Answer 1

Part a: The value of Y_A and Y_B are
`\frac{\bar{u}_A}{x}` and
`100-x-\frac{\bar{u}_A}{x}` respectively.

Part b: Y_A and Y_B are given as
`\frac{\bar{u}_A}{50}` and
`50-\frac{\bar{u}_A}{50}` 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

Step-by-step explanation:

Part a

For the value of the utility function is given as

`\bar{u}_A=xY_A\\Y_A=\frac{\bar{u}_A}{x}`

Also the YB is given as

`Y_A+Y_B=100-x\\Y_B=100-x-Y_A\\Y_B=100-x-\frac{\bar{u}_A}{x}`

So the value of Y_A and Y_B are
`\frac{\bar{u}_A}{x}` and
`100-x-\frac{\bar{u}_A}{x}` respectively.

Part b:

Now

`\bar{u}_B=xY_B\\\bar{u}_B=x(100-x-\frac{\bar{u}_A}{x})\\\bar{u}_B=100x-x^2-\bar{u}_A`

For the maximization

`\frac{\partial \bar{u}_B}{\partial x}=0\\\frac{\partial (100x-x^2-\bar{u}_A)}{\partial x}=0\\100-2x=0\\x=100/2\\x=50`

From question 1 Y_A and Y_B are given as
`\frac{\bar{u}_A}{50}` and
`50-\frac{\bar{u}_A}{50}` respectively for maximization of Y_B

Part c:

At the Pareto efficient allocation

`(Mu_X)/(Mu_(Y_A))=(Mu_X)/(Mu_(Y_B))`

This is simplified to

`(Y_A)/(x)=(Y_B)/(x)\\Y_A=Y_B`

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