Consider a collective decision by three individuals to produce, or not, one public good that costs $150. Suppose that if the public good is produced, the cost is equally shared among the three individuals, namely $50 each. Assume that the gross benefits from the public good differ among individuals and are respectively $20, $40, and $100 for individuals 1, 2, and 3. Each individual is asked to announce his own benefit for the public good, and the public good is produced only if the sum of reported benefits exceeds the total cost. 

a. Show that the Groves-Clarke tax induces truth-telling as a dominant strategy if each individual reports independently his own benefit. 

b. Show that the resulting provision of public good is optimal. 

c. Show that the Groves-Clarke tax is not robust to collusion in the sense that two individuals could be better off by jointly misreporting their benefit from the public good. 

d. What would be the provision of public good if the decision were taken by a majority vote, assuming that the cost is equally shared in the event of public good provision? Compare your answer with part b, and interpret the difference.