Tell It by Myself…

The Lindahl Equilibrium focuses on how to maximize the people’s welfare, like the Samuelson Condition, however, they have little differences.

We assume there are only 2 people in the society, here is Smith and Jones.

As an example for Smith, the demand for the Quantity of public good will decrease since the share of cost increase (No one want to pay too much)

The intersection point, which is $C$ , is the Lindahl Equilibrium for Smith and Jones

The Numerical Path

Also consider in a 2 player model, the utility function for player 1 is

u_{1} = x_{1} G^{α}

For player 2, it is

u_{2} = x_{2} G

Notice that $G = g_{1} + g_{2}$ and $x_{i}$ denotes to the players’ private consumption, so

x_{1} + g_{1} = x_{2} + g_{2} = 1

We want to maximize:

ma x u_{i} (M_{i} - τ_{i} pG, G)

The $M$ is the endowment, $M_{i} - τ_{i} pG$ is the $x$ (private goods) from my point of view.

Then we have to come up with the FOC, that is

τ_{i} p = \frac{u _{G}^{i}}{u _{X}^{i}}

And this is the most important part. Also remember $\sum_{i = 1}^{n} τ_{i} = 1$