# The Theoretical Analysis of Public Goods: The Lindahl Equilibrium

## The Lindahl Equilibrium

We now investigate the Lindahl equilibrium whereby the government plays an important role in the provision of public goods. See Lindahl (1919). This equilibrium can attain the efficiency of allocation. Thus, many studies have considered the plausibility of this mechanism. The mechanism has three stages.

• (i) The government determines each agent’s burden ratio of the provision of public goods.
• (ii) Each agent declares her or his desirable provision of public goods at a person- specific burden ratio.
• (iii) The government adjusts burden ratios so as to equate all agents’ desirable levels in public goods. Then, it determines the optimal provision of public goods whereby all agents demand the same level of public goods.

First, let us formulate person 1’s optimizing behavior. She or he maximizes her or his utility, Eq. (11.13), subject to the following budget constraint:

where h = y1/Y is the person-specific burden ratio for person 1. h is given for person 1 and ph may be regarded as the effective price or personalized price of a public good.

As shown in Fig. 11.6, the slope of the budget line AB is ph and OA = M. The optimal point is given as point E, where the budget line and indifference curve are tangent. The associated Y is person 1’s most desirable level of public goods.

The relation between h and person 1’s optimal Y1 may be written as

This is called the Lindahl reaction function. Y1 decreases with h. If the personalized price of a public good, h, increases, person 1’s demand for Y1 declines because of substitution and income effects.

Fig. 11.6 The optimizing behavior

Person 2’s optimizing behavior may be formulated in a similar way:

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Note that person 2’s burden ratio is given as 1 — h. The government presents a person-specific burden ratio to each agent. Usually, two persons’ optimal levels of public goods are not the same. Then, the government adjusts the burden ratios so that the sum of the burden ratios becomes 1 at the same optimal level of Y, Y1 = Y2.