[BONUS, 16 points] Suppose that there are fifteen students 1, 3, 5, 7,..., 29 ea

Question

[BONUS, 16 points] Suppose that there are fifteen students 1, 3, 5, 7,..., 29 each holding a single gold bar and fifteen students 2, 4, 6, ..., 30 just have money, and no gold. A student only needs one gold bar, and twice her student number is what that one bar is worth to her as a buyer, while three times her student number is the cost as a seller. Additional bars are useless. So odd students are the potential sellers, and even students are potential buyers. Into this mix walks Professor Q with a box full of 10 gold bars that she has no use for at all (i.e., are worthless).

a) [6 points] Find the market-clearing price range and quantity without Professor Q,

b) [4points] Find the new market clearing price and quantity with Professor Q
c) [3 points] How much did Professor Q depress the price of the bars of gold?
d) [3 points] How much did Professor Q increase the quantity of bars of go

Explanation / Answer

Given that the worth of one bar to a student is twice her student number which denotes the reservation price or willingness to pay. So a student with number 14 is likely to pay $28 for 1 gold bar.

The cost is three times her student number which implies a student numbered 11 has a cost of $33 for 1 gold bar. Additional bars are useless. So odd students are the potential sellers, and even students are potential buyers.

a) A market clearing price should be the one that equates the marginal willingness to pay with the cost so that at the equilibrium, what the seller is willing to charge is what the buyer in willing to pay. Note that the first seller is willing to sell the gold bar at $3 while the maximum willingness to pay is $60 by 30th student. If we think like this, the student number 30, will get the first bar, student with number 28 getting the second and it goes on till student with number 20 with a willingness to buy at $40 getting a gold from a student with number 11 having a cost worth $33 per bar. Beyond this, the seller with number 13 demands $39 for a gold bar while the maximum willingness is $36 by student number 18. So no trade occurs after this point

The price is thus $40 at which the demand and supply are closest to the equilibrium. The quantity sold at this point seems to be 6 gold bar.

b) When Professor Q comes with a box full of 10 gold bars that she has no use for at all, this would mean 10 bars will be sold immediately with 10 buyers, in order of decreasing willingness to pay. So student with number 30 to 12 are able to get one gold bar without spending. Student with number 10 has a willingness to pay worth $20 per gold bar so she buys one with seller number 1. Student with number 8 has a willingness to pay worth $16 per gold bar so she buys one with seller number 3 with cost $9. The student with number 6 will not be able to buy any gold since the price charged by seller number 5 is $15.

So new market clearing price is $16 and quantity with Professor Q is 12 gold bars.

c) Professor Q depress the price of the bars of gold from $40 to $16 per gold bar.
d) Professor Q increase the quantity of bars of gold bar from 6 to 12 gold bars.

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