Suppose the government announces a prize for a solution to its need. Both firm 1

Question

Suppose the government announces a prize for a solution to its need. Both firm 1 and firm 2 have an idea for a technology to satisfy the government’s need. Firm 1’s idea is (1, 1) = (8,10) and firm 2’s idea is (2, 2) = (7, 2), where is the per-period consumer surplus of the technology if competitively supplied, and is the cost of developing the idea into a technology. Interest rate is 10%. The government wants to hold a Vickery auction to pick the best idea.

If you are the CEO of firm 1 and someone you trust tells you that firm 2 will report a social value of $55. What is your profit if you win the bid? If you lose the bid?

Explanation / Answer

(The interpretation is that a % b as long as the utility improvement of b over a is

“imperceptible”.) Must this preference relation be

(a) reflexive,

(b) complete,

(c) transitive?

The preference relation % defines the corresponding “strictly preferred” relation

and “indifference” relation as follows:

a b [a % b] ¬[b % a] (1.2)

a b [a % b] [b % a] (1.3)

(d) Must the strictly preferred relation be transitive?

(e) Must the indifference relation be transitive?

Suppose now that we are instead given a complete preference relation % over a finite

set X whose corresponding “strictly preferred” relation is transitive. Can we always

find a utility function u : X 7 R such that % is represented by (1.1)? Prove or

disprove with a counterexample, distinguishing between two cases:

(f) X has three elements,

(g) X has more than three elements.

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