# Pareto Efficient level of Public Good

As part of self study towards an entrance exam, I am solving the following question.

There are two consumers in the economy and two goods, one private and one public. The utilities of the consumers is given by:

$$u_1(x_1,y) = x_1+6\sqrt y$$

$$u_2(x_2,y = x_2 + 10\sqrt y$$

where $$x_i$$ is the quantity of the private good and $$y$$ is the quantity of the public good consumed. Initial endowments of the private goods:

$$\omega_1 = 40, \omega_2 = 60$$.

A unit of private good can be converted one-for-one to a public good.

My attempt:

The Pareto Efficient level of public good is determined by the equation:

$$|MRS_1| + |MRS_2| = MC(G) \quad (1)$$

where $$MRS_1 = \frac{MU_G}{MU_{x_1}}$$, $$MRS_2 = \frac{MU_G}{MU_{x_2}}$$ and $$MC(G)$$ is the marginal cost of providing the public good.

Solving $$(1)$$ with the given utility functions I get:

$$\frac{3}{\sqrt y} + \frac{5}{\sqrt y} = 1$$ $$\implies y = 64$$

Therefore, the Pareto efficient level of public good is $$y=64$$.

The question asks to select the Pareto Inefficient bundle $$(x_1,x_2,y)$$ amongst the following:

A. $$(50,0,50)$$

B. $$(90,0,10)$$

C. $$(0,80,20)$$

D. $$(16,20,64)$$

I think all of A, B and C are Pareto Inefficient.

Option C is not Pareto efficient because a Pareto improvement exists. At $$(0,80,20)$$, $$\begin{equation} u_1(0,20)=0+6\sqrt{20}\approx 26.83 \qquad u_2(80,20)=80+10\sqrt{20}\approx124.72. \end{equation}$$ Observe that at this bundle, consumer 2's MU for the good $$y$$ is $$5/\sqrt{20}\approx1.11$$ and his MU for $$x$$ is $$1$$. Therefore, increasing one unit of good $$y$$ at the expense of one less unit of $$x$$ is utility enhancing for consumer 2. At the same time, one extra $$y$$ benefits consumer 1 as well. So we can easily verify that $$(0,79,21)$$ is a Pareto improvement over $$(0,80,20)$$: $$\begin{equation} u_1(0,21)=0+6\sqrt{21}\approx27.495 \qquad u_2(79,21)=79+10\sqrt{21}\approx124.83 \end{equation}$$ where both consumers experience higher utilities.