The Joy of International Trade
By John P. Robertson
Some of the following questions do not have a unique answer, but I’ve had so much fun playing with them that I think you might too. Jon Evans suggested the puzzle.
Countries A and B both produce and consume only televisions (TVs) and recreational vehicles (RVs). The happiness in each country is the number of TVs consumed times the number of RVs consumed in that country. For example, 1,000 TVs and 4,000 RVs provide the same happiness as 40,000 TVs and 100 RVs. A country must consume some TVs and some RVs, or there is no happiness in the land. However, neither country cares about the happiness of the other country.
Country A can produce 10,000 TVs or 2,000 RVs or any linear combination, 10,000a TVs and 2,000(1 - a) RVs for 0 ≤ a ≤ 1. Country B can produce 100,000 TVs or 10,000 RVs or any linear combination, 100,000b TVs and 10,000(1 - b) RVs for 0 ≤ b ≤ 1.
Here are the questions:
Liars, Truth Tellers, and Random Answers
- If there is no trade between the countries, what is the optimal production and consumption in each country, and what is the total happiness in each country?
- Show that with trade the happiness in both countries can be greater than without trade.
- If A and B agree to trade under the rule that average prices of all goods exchanged between them are equal to their final market prices, what is the maximum happiness in A and B under optimal production, consumption, and trade?
- Under trade, but without the exchange price constraint in (c), what do you think will happen? How much will each country produce, trade, consume? How happy will each country be? Or, what is the range of possibilities?
The puzzle was that you are at a fork in the road and you want to know which of two roads leads to the village. Three natives are present, one who always tells the truth, one who always lies, and one who answers at random, but you don’t know who is who. How can you ask two yes-or-no questions, each question addressed to one particular native, and determine which road leads to the village?
David Uhland’s solution uses the first question to make it possible to direct the second question to a native that you know is not the random answerer. David’s solution is to ask the first person, “If I were to ask you ‘Does the second person give random answers?’, would you say ‘yes’?” Either the first person is the random answerer or the answer reveals the random answerer’s position. You direct the second question to whichever of the second and third persons is indicated to not be the random answerer. The second question could be: “If I were to ask you ‘Does the path on the left lead to the village?’ would you say ‘yes’?” Whether you ask the truth teller or the liar (but not the random answerer, who has been eliminated), a “yes” answer indicates that the left path leads to the village and a “no” answer means the path to the right leads to the village.
Christopher Allard, Russell Fisher, Alexander Kozmin, Ben Kraus, and Jason Russ also submitted solutions.