Finding the Pareto efficient allocations

Question

Consider a 2 person 2 good economy where there is a private good $x$ and a public good $y$. Agent 1 has an endowment of 10 units of the private good and Agent 2 has an endowment of 20 units of the private good. Initially, there is no public good in the economy. In order to produce $y$ units of the public good, $y^2$ units of the private good should be used. That is, the cost function is $c(y)=y^2$.

Utility functions of the agents are as follows;

$$u_1(x_1,y)=x_1+y$$ $$u_2(x_2,y)=x_2y$$

Firstly, I need to find the Pareto efficient allocations where 4 units of public good are produced.

Secondly, I need to find the private consumption of both agents as well as the public good level at the equilibrium.

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In other to find the Pareto efficient allocations, I maximize the sum of payoffs of both agents

$$max\{ u_1(x_1,y)+ u_2(x_2,y)-c(y)\}$$

First order condition with respect to $y$ is $6+x_2-2y=0$

So, I found $x_2=2\times 4-6=2$

But my attempt is not true. I cannot do the correct solution. I will appreciate if you help me to solve the parts of this question.

Answer 1

Set of feasible allocations in this economy is:

$\mathcal{F}=\{(x_1,x_2,y)\in\mathbb{R}^3_+|x_1+x_2+y^2=30\}$

This set can also be represented in graph in the following way:

To determine (interior) efficient allocations, we can first calculate MRS of individual 2 whose utility is $u_2(x_2,y)=x_2y$, and get:

$\text{MRS}_2 = \frac{y}{x_2}$

Now we can rewrite $u_1(x_1, y)=x_1+y$ as $u_1^{\text{ADJ}}(x_2,y) = u_1(30-x_2-y^2, y)=30-x_2-y^2 + y$. We have just written $u_1^{\text{ADJ}}$ as a function of $x_2$ and $y$, by taking into account the feasibility constraint. We now calculate MRS of individual 1 using $u_1^{\text{ADJ}}(x_2,y)$, and get:

$\text{MRS}_1^{\text{ADJ}} = \frac{1}{2y-1}$

(Interior) efficient allocations satisfy the condition:

$\text{MRS}_1^{\text{ADJ}}=\text{MRS}_2$ which gives: $x_2 = 2y^2-y$

Here is the set of efficient allocations in the graph:

Clearly, there is no (interior) feasible Pareto efficient allocation that satisfy $y=4$. We can also check this by looking at the condition $x_2 = 2y^2-y$ which yields $x_2=28$ when we plug in $y=4$, and that is not feasible.

To show that there is no efficient allocation at which $y=4$ holds at the edges also, we just need to consider two allocations:

• $(x_1,x_2,y)=(0,14,4)$. Observe that $(x_1,x_2,y)=(1,20,3)$ is Pareto Superior.
• $(x_1,x_2,y)=(14,0,4)$. Observe that $(x_1,x_2,y)=(21,0,3)$ is Pareto Superior.

Therefore, there is no Pareto efficient allocation where 4 units of public good are produced.

Answer 2

Let $y_i$ be the contribution of agent i towards the production of $y$

$y=\sqrt{\sum y_i}$

So when $y=4$ $\Rightarrow \sum y_i=16 \Rightarrow \sum x_i = 14$

The set of pareto efficient allocations is $\{(x_1,x_2):x_1+x_2=14,x_1 \in [0,10], x_2 \in [0,20]\}$

Private consumption of the goods will be the same as the endowments and the provision of public good will be $0$ at the equilibrium.

Reason:

$max_{{x_1,y_1}: x_1 + y_1 = 10} x_1 + \sqrt{\sum y_i}$

$\Rightarrow \frac{39}{4} + y_{2} = x_{1}$

$max_{(x_2, y_2):x_2+y_2=20} x_2 \sqrt{\sum y_i}$

$\Rightarrow x_2= \frac{1}{3}(2y_1 + 40)$

Let the price of the private good be $1$. Given the utility function of agent $1$ this implies $y_1=y \Rightarrow y_2=y^2 - y$ Clearly the above system of equations has no real solution. So equilibrium does not exist.

Edit: The equilibrium will be when $x_1=10, y_1=0, x_2=\frac{40}{3},y_2=\frac{20}{3}$ Credits: Agrim Rana