Microeconomics · Hard · 30 marks
IEO past paper: 2026 International Economics Olympiad, open questions
Open Question 4. Utility maximization, social preferences, and joint contributions
Anil has won £100 in the lottery and is deciding how to split it between himself (x, the amount he keeps) and his friend Bala (y, the amount he gives). Suppose Anil has the utility function u ( x , y ) = x − δ ( y − x ) 2 u(x, y) = x - \delta (y - x)^2 u ( x , y ) = x − δ ( y − x ) 2 δ ≥ 0 \delta \ge 0 δ ≥ 0
(a) Suppose you observe Anil giving £37.5 (out of £100) to Bala and keeping the rest for himself. What value of δ is consistent with this behaviour? [6] marks (b) How does the amount Anil gives to Bala vary with δ? Refer to Anil's utility function when explaining your answer. (You can give a mathematical and/or an intuitive explanation.) [6] marks (c) Now suppose that Anil becomes purely selfish and plays a one-round game with Bala, Cindy and David (4 people total).
- Each player is given £20 and chooses a contribution to a common pot.
- At the end of the round, every player receives k = 0.2 times the total amount in the pot plus the money they chose not to contribute.
- If a player contributes less than some expected amount C, they are punished: an amount P deducted from their money at the end of the round.
Suppose P = £6. What is the largest minimum contribution that Anil would be willing to make (that is, find maximum C that he will agree to contribute)? Assume that he meets the requirement when indifferent. [6] marks (d) If the share of the total amount in the pot changes to k = 0.5, what is the largest minimum contribution C Anil would be willing to contribute if the punishment is P = £5? [6] marks (e) Before the round, the four players jointly agree on a single punishment size P and a single required minimum contribution C that will apply to all of them. Once play begins each player is again purely selfish. Anticipating this behaviour, the group picks P and C to maximise the combined end-of-round wealth of all four players. If indifferent between different levels of P, they choose the smallest. Determine the P and C the group should agree on when (i) k = 0.2 and (ii) k = 0.5, and give the resulting combined wealth in each case. [6] marks
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