# Finding the Pareto efficient allocations

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.

• Hi! 1. What textbook did you read, what mathematical condition are you using to determine Pareto-efficiency? 2. Why do you think your attempt is incorrect? If you are sure it is incorrect, why include it? Aug 14, 2022 at 22:20
• I remember this definition. I have an answer key but it only includes its results at the end. And my result doesn’t coincide with that of the answer key. I included my solution because when I post the question without any attempt, my post is either closed or downvoted. Due to rules, I add my wrong answer. @Giskard Aug 14, 2022 at 23:18
• @studentp Are/Were you interested in the Lindahl equilibrium or competitive equilibrium?
– Amit
Apr 13 at 3:58

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.

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

• I see. Thank you for your great answer! Aug 15, 2022 at 10:38