Question

Suppose we have a set of three buyers who are trying to buy homes from three sellers. Suppose that seller A charges 3 for their house, seller B charges 2 for their house, and seller C charges 1 for their house The valuations for the three buyers are as follows ordered by house number (so house a, then house b, then house c):

Buyer 1: A: 1 B: 3 C: 2
Buyer 2: A: 5 B:5 C: 5
Buyer 3: A: 5 B: 3 C: 2
Assuming that this set of prices is market clearing, provide the utility for each player in the transaction.

Answer 1

As we can see from the following question that :

`$\text{Seller A}$` charges
`$3$` for their house,
`$C_A=3$`

`$\text{Seller B}$` charges
`$2$` for their house,
`$C_B=2$`

`$\text{Seller C}$` charges
`$1$` for their house,
`$C_C=1$`

Now depending on the valuation that each buyer have on these three houses, they derive utility from it :

Buyer I

Valuation for house A,
`$V'_A=1$`

Valuation for house B,
`$V'_B=3$`

Valuation for house C,
`$V'_C=2$`

Utility that buyer I derives from :

House A,
`$V_A = V'_A-C_A = 1-3 = -2$`

House B,
`$V_B = V'_B-C_B = 3-2 = 1$`

House C,
`$V_C = V'_C-C_C = 2-1 = 1$`

As we can see from buying house B and C, buyer I has positively utility, so the buyer will be interested to buy house B and C. As the utility derived from both the houses is same, he will be indifferent.

Buyer II

Valuation for house A,
`$V^2_A=5$`

Valuation for house B,
`$V^2_B=5$`

Valuation for house C,
`$V^2_C=5$`

Utility that buyer II derives from :

House A,
`$V_A = V^2_A-C_A = 5-3 = 2$`

House B,
`$V_B = V^2_B-C_B = 5-2 = 3$`

House C,
`$V_C = V^2_C-C_C = 5-1 = 4$`

Buyer II drives positive utility from all the three houses, so he will be interested to buy any of them but will be inclined towards house C, as it gives the buyer more utility.

Buyer III

Valuation for house A,
`$V^2_A=5$`

Valuation for house B,
`$V^2_B=3$`

Valuation for house C,
`$V^2_C=2$`

Utility that buyer III derives from :

House A,
`$V_A = V^2_A-C_A = 5-3 = 2$`

House B,
`$V_B = V^2_B-C_B = 3-2 = 1$`

House C,
`$V_C = V^2_C-C_C = 2-1 = 1$`

The buyer III derives positive utility from all the three houses but will be more inclined to buy house A as it gives the buyer higher utility as compared to the other houses.