Lindahl Tax and Pareto Optimality
A very important question is that whether a Lindahl tax is a Pareto Optimal equilibrium? A Pareto Optimal allocation happens with public goods when the total of the marginal rates of substitution (MRS) equals the marginal rate of transformation (MRT). So if it can be shown that this holds true in a Lindahl equilibrium, it can be conveniently said that it is Pareto Optimal. This can be shown by following the following steps:
We take a demand curve for a public good. The less the price of the public good, the more will X want to consume. Let the horizontal line (dashed) be the full price of the public good. Now here, the demand curve implies that X will demand very less. But what if instead of the price decreasing, the percentage of the price X has to pay decreases? Now X sees the price going down, so his demand for the good increases. Now lets consider the demand curve of another person, lets say Y. Y sees the vertical axis turned the other way around, with the full price on the bottom and percentage decreasing as you move upwards. Like X, Y will also demand more as his observed price goes down.
Now as Y observes the price going down it also means that we move further up the vertical axis. Equilibrium is when both X and Y demand the equal amount of the good. This is possible only when the demand curves of both X and Y intersect each other. If a line is drawn over the price axis from that point of intersection, we get the percentage share for each person that is required to get that price.
In the Lindahl tax scheme it is essential that the system should provide for a pareto optimal output of the public good. The other important condition is that the Lindahl tax scheme should connect the tax paid by an individual to the benefits he derives. This system kind of promotes justice. If the individual's tax payment is equivalent to the benefits received by him, and if this linkage is good enough then it leads to Pareto Optimality.
So it is observed that X is paying P*45% per unit, and Y is paying P*55% per unit, and the economy produces Q* units. This point is called the Lindahl Equilibrium, and the corresponding prices are called Lindahl prices.
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