Groves-Ledyard Mechanism: Concept & Explanation

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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2. Role of Government in Economy
3. Role of Government in Mixed Economy: Public & Private Sector
4. Role of Government under Cooperation and Competition
5. Role of Government in Economic Development and Planning
6. Concept of Public Goods, Private Goods, and Merit Goods
7. Concept of Market Failure and Functions of Government
8. Market Failure and Functions of Government: Decreasing Costs
9. Market Failure and Functions of Government: Externalities
10. Market Failure and Functions of Government: Public Goods
11. Future Market: Meaning, Role & Uncertainty
12. Concept of Information Asymmetry
13. Theory of Second Best: Concept & Explanation
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15. Preferences: Meaning, Types & Problems of Preference Revelation
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18. Economic Theory of Democracy: Concept & Explanation
19. Politico Eco Bureaucracy: Concept & Explanation
20. Rent-Seeking and Directly Unproductive Profit-Seeking Activities
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22. Benefit Theory or Voluntary Exchange Theory
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24. Bowen Model: Concept, Advantages & Limitations
25. Samuelson’s Model of Public Expenditure
26. Musgrave’s Model of Public Expenditures
27. Demand Revealing Schemes for Public Goods
28. Vickery-Clarke-Groves Mechanism
29. Groves-Ledyard Mechanism
30. Tiebout Model: Concept, Assumptions Equilibrium & Simple Tiebout Model
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32. Keynesian Principles of Stabilization Policy
33. Difference Between Keynesian Economic Thought and Others
34. Role of Expectations and Uncertainty in Formulating Stabilization Policy
35. Intertemporal Markets Efficiency & Failure
36. Liquidity Preference Theory
37. Diamond-Dybvig Banking Model
38. Preference Shocks, Adverse Selection & Central Bank
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51. Performance Based Budgeting and Zero Based Budgeting
52. Theories of Tax Incidence: Concentration Theory, Diffusion Theory & Modern Theory
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54. Equity Principle and Efficiency Principle of Taxation: Meaning, Explanation & Examples
55. Ability to Pay and Benefits Received Principle of Taxation
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57. Deadweight Loss of Taxation: Causes, Measurement & Example
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62. Solution to Problem of Double Taxation: Intra-Country & International
63. Double Taxation Avoidance Agreement (DTAA) and Indian Policy
64. Classical View on Public Debt
65. Compensatory Aspect of Public Debt Policy
66. Public Debt or Borrowings: Concept, Need, Sources & Types
67. Concept of Public Debt or Public Borrowings
68. Need for Public Debt or Public Borrowing
69. Sources of Public Debt
70. Classification of Public Debt
71. Burden of Public Debt: Meaning, Types & Explanation
72. Debt Through Created Money or Deficit Financing
73. Public Debt (Public Borrowings) and Inflation (Price Level)
74. Crowding Out of Private Investment and Activity
75. Principle of Public Debt Management and Debt Repayment