Groves and Ledyard jointly worked hard and gave the demand-revealing mechanism which they called by the name “An optimal government”. This mechanism is used to formulate the rules of a game in which the amount of taxes and distribution of public goods is being determined by the government as a result of those messages which the individuals choose to communicate.
Though the government has no independent knowledge about the preferences and the individuals are aware that giving a signal of their preferences to the government will be beneficial for the government and it might be possible that Pareto optimal situation in the economy.
The Groves-Ledyard mechanism is defined for general equilibrium and applies to smooth preferences. They discovered this scheme in 1975 and worked out this mechanism in a quasi-linear form.
Vx (Ax, B) = Ax + Fx(B) (Eq 1.1)
The advantage of the Clarke tax is that there is a dominant strategy for the equilibrium, but the disadvantage is that though this tax system was leading to a Pareto efficient situation but some amount of private goods are being wasted in this mechanism.
Suppose there are x consumers, and there is one public good and one private good. Each consumer has an initial endowment of Vx units of private goods. Public goods are being produced at a constant unit cost of q. The government will ask each consumer x to submit a number of (positive or negative) mx. The government will supply an amount of public good B=∑xmx.
Now, the Groves-Ledyard mechanism is presented as:
─mx = 1/ n-1 ∑y≠1my (Eq 1.2)
To be the average of the numbers submitted by persons other than x. The function can also be defined as:
Rx(n) = 1/n-2 ∑y≠x (my-⌐ m- x) 2 (Eq 1.3)
The thing that is being noticed in equation 1.3 is that Rx (n) depends on the –m-x for y≠x but does not depend on my. These expressions will help in making a balanced budget when vector messages sent by individual consumers are m= (m1─mn). The Groves-Ledyard mechanism will impose a tax on individual x equal.
Tx(m) = αxq ∑nk=1 mk+¥/2 (n/n-1)(mx- mx) 2 (Eq 1.4)
Where αx and ¥ are arbitrarily chosen positive parameters and ∑k αx =1
If the vector message is m = (m1- mn), then consumer x utility will be
Vx – Tx(m) + Fx (∑nk=1 mk) (Eq 1.5)
In the case of Nash equilibrium, each consumer x would be choosing mx to maximize in equation 1.5. The first order for maximizing is:
Fx (∑k mk) = ¥ [mx – 1/n ∑k mk] + αx q (Eq 1.6)
Now, summing up the equation ∑kαk = 1 and ∑k Fk (∑k mk ) = q
This is Samuelsson’s condition for the efficient provision of public goods. The tricky thing to depict is that the total revenue collected by the Groves Ledyard mechanism is equal to the cost of the public good. Thus, to find out the total of the sum of the taxes collected from each consumer x then it is being found out that
∑nx=1 Tx (m= ∑nx=1 αx q ∑nk=1 mk + ¥/2 ∑nx=1(n/n-1(mx- m- x)2 – Rx (m) (Eq 1.7)
Now, fiddling with the sum of quadratic, the result will be
∑nx=1 n/n-1 (mx – m- x)2 = ∑nx=1 Rx (m) (Eq 1.8)
Therefore, equation 1.8 can be simplified as follows
∑nx =1 Tx (m) = ∑nx =1 αx q ∑nk =1 mk (Eq 1.9)
Since ∑nk =1 and mx = B and ∑nx =1 αx = 1 this equation is further simplified as follows
∑nx =1 Tx (m) = q (B)
This means that revenue is exactly covering the cost of the public good.
Now, Groves Ledyard Mechanism is presented in quasi-linear utility form:
It is of great interest to analyze the Groves Ledyard Mechanism that the nature of this mechanism is applied in quasi-linear utility where each consumer x has a utility function. Vx (Ax B) = Ax + Fx (B). This type of model gives a unique solution, as given by Clarke. This type of equilibrium can be easily computed and described.
Since Fk < 0, then equation 1.1 has a unique solution for ∑kmk. Let B denote this solution k, and now this equation can be defined as follows.
Bx = Fx (B)
Now αx q and ¥ are parameters, and Bx is uniquely solved. Thus, the unique solution for mx is as follows:
Mx= 1/ ¥ (Bx – αx q) B/n
Thus, this is the quasi-linear explanation of the Groves-Ledyard mechanism.
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