Business: 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

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Public Good. Suppose there are two goods X and Y in

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20.01.2022

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

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

Explanation:

Part a

For the value of the utility function is given as

$\bar{u}_A=xY_A\\Y_A=\dfrac{\bar{u}_A}{x}$

Also the YB is given as

$Y_A+Y_B=100-x\\Y_B=100-x-Y_A\\Y_B=100-x-\dfrac{\bar{u}_A}{x}$

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

Part b:

Now

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

For the maximization

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

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

Part c:

At the Pareto efficient allocation

$\dfrac{Mu_X}{Mu_{Y_A}}=\dfrac{Mu_X}{Mu_{Y_B}}$

This is simplified to

$\dfrac{Y_A}{x}=\dfrac{Y_B}{x}\\Y_A=Y_B$

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

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