Problem
Endowment equilibrium: Farmers and miners
Suppose an economy consists of 2 people: A farmer and a miner. The two individuals have identical preferences, u (f, g) = f g, where f is a kilo of food, and g is a gram of gold. The farmer is endowed with 20 kilos of food and no gold. The miner is endowed with no food and one gram of gold. The farmer and the miner can trade food and gold at prices pf and pg. In the following, choose food to be the numeraire good, so set pf = 1 and equilibrium will determine the price of gold pg.
I. For given prices (pf = 1 and pg), solve for the farmer's optimal demand for food and gold given that the farmer has an endowment that has value 20.
II. For given prices (pf = 1 and pg), solve for the miner's optimal demand for food and gold given that the miner has an endowment that has value 1 ×pg.
III. Determine the equilibrium price of gold, p∗g. How does the value of the miner's endowment compare to that of the farmer?
IV. Illustrate the equilibrium outcome in an Edgeworth Box.
V. Determine the contract curve for the economy.
VI. Illustrate the contract curve in the same Edgeworth Box.
VII. Now, suppose 8 more farmers identical to the first enter the economy. Determine the endowment equilibrium for the economy with 9 farmers and 1 miner. The 9 farmers each have an endowment of 20 kilos of food and no gold. The miner has a gram of gold and no food. Keep that food is the numeraire good, pf = 1.
i. What is the equilibrium consumption bundle of a farmer?
ii. What is the equilibrium consumption bundle of the miner?
iii. What is the equilibrium price of gold?
iv. What happened to the value of the miner's endowment upon the entry of more farmers? Explain.
VIII. Finally, let us extend our insights into the area of occupational choice. That is, what would happen if each individual had the choice to become either a farmer or a miner? Specifically, suppose the economy consists of 10 individuals. Let everything else be the same as above: Preferences are the same. Upon "graduating" from occupational training, a miner is endowed with 1 gram of gold and no food, and a farmer is endowed with 20 kilos of food and no gold. We need to come up with reasonable equilibrium concept for this economy. Foreshadowing, we will think of an equilibrium where no individual has "regrets" upon seeing the equilibrium outcome of the economy. When we get to the game theory module of the course, this will be associated with the Nash equilibrium concept. For now, simply say that the economy will 1 have n farmers and m miners such that n +m = 10. "No regrets" translate to the requirement that in equilibrium: (i) No farmer would prefer to have become a miner, and (ii) No miner would have preferred to become a farmer. I want you to arrive at a prediction about how many miners and farmers the economy will have along with the overall production of food and gold. For this, the following provides an analysis roadmap:
i. Consider an economy with n farmers and m miners. Using your expressions for farmer and miner consumption choices from 7(a) and 7(b) determine the equilibrium price of gold.
ii. For a given occupation allocation (m, n), what is the equilibrium utility of any given farmer, u∗f (m, n)? What is the equilibrium utility of any given miner u∗m(m, n)?
iii. In equilibrium, no regrets means that no individual will want to "deviate" from their chosen occupation. So, equilibrium requires that u∗f (m, n) ≥u∗f (m + 1, n -1) that is, a farmer asks, would I better off if I chose to be a miner and the economy has one less farmer and one more miner? Same kind of consideration by a miner imposes the equilibrium constraint, u∗m (m, n) ≥u∗f (m -1, n + 1). I will make our lives a little easier and ask you for a simplified equilibrium requirement: Find the equilibrium occupational allocation (m∗, n∗) such that u∗f (m∗, n∗) = u∗m (m∗, n∗) and m∗+ n∗ = 10. The simplification I am using is based on thinking about large populations where a single individual's decision to change occupation does not impact the overall equilibrium in any significant way. That is not true when the economy has only 10 individuals, but I would like to abstract from this "annoyance" for the time being, and I can reveal that the equilibrium we will find using this approach will nevertheless be a valid equilibrium in our setting with 10 individuals.
iv. Given the equilibrium you found in (c), what is the economy's total food and gold production? What is the price of gold?