Lindahl Tax - Mathematical Representation

Mathematical Representation

We assume that there are two goods in a n economy:the first one is a "public good," and the second is “everything else.” The price of the public good can be assumed to be Ppublic and the price of everything else can be Pelse.

  • α*P(PUBLIC)/P(EVERY) = MRS(PERSON1)

This is just the usual price ratio/marginal rate of substitution deal the only change is that we multiply Ppublic by α to allow for the price adjustment to the public good. Similarly, Person 2 will choose his bundle such that:

  • (1-ɑ)*P(PUBLIC)/P(EVERY)= MRS(PERSON2)

Now we have both individuals' utility maximizing. We know that in a competitive equilibrium, the marginal cost ratio (price ratio)should be equal to the marginal rate of transformation, or

  • MC(PUBLIC)/MC(EVERY)==MRT

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