Samuelson Condition

micro

Tell It by Myself…

The utility function:

$u_i=x_{i}G$ , where $G=\sum g_i$ and $x_i+g_i=1$

For the person himself

His constraint would be $x_i+p{g}_i=M_i$ and $M_i$ would be his Endowment, you can consider in the all the resource he has

Here , $x_i$ maybe the private part of the individual, like the one in the Lindahl Equilibrium. $g_1$ represents the public part by the individual, that means,

Understanding

Since $x_i$ is all the private part, so you don’t need to multiply anything, what you have is what you consume. But $p$ here is the price of the public goods and you should multiply $g_i$ and $g_i$ is what you consume for the public goods.

To reach the optimal:

We should consider in the Pareto Optimality so we need to use the Largrangian Multiplier in order to get it.

The Utility Equation

$$u_i(x_i,g_i+\sum _{j\ne i}{g}_j)$$

You will find that, $g_i+{\sum }_{j\ne i}{g}_j$ is just the sum up of $G$ , so I think it can also be represented by

$$u_i(x_i,G)$$

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Reference