Housing prices in 2021 and how it applies to Networks
The housing market in the United States has been subject to historically high divides between housing prices and median household income especially as a result of the pandemic. As more people lost their jobs as a result of COVID-driven unemployment, the average house price to income ratio has risen drastically. From 2019 to 2021, this ratio increased from 4.7 to 5.4, in other words, “homes cost 5.4x what the average person earns in one year”.
The increase in average home values has seen an even starker change when compared to 1965. In the latter, the average home value was around $171,942. However, this number has increased by 118%, where now the average home value is at around $374,900. Yet, median household income has only increased 15% in those 56 years, from $59,920 to $69,178 (in 2021 inflation-adjusted dollars). Therefore, since 1965, “home prices have increased 7.6x faster than income”.
This disparity provides a clear parallel to the topic of matchings learnt in class. Matchings entail a set of edges (in a bipartite graph) that do not share common vertices. A perfect matching requires that all nodes from the bipartite graph contain a matching. However, if there isn’t a perfect matching, there must be a constricted set (matching theorem). This means that there either is a shortage in supply for a certain good (high demand) or there is a surplus of a good (low demand). I believe that in this case, there is a constricted set in the way that no person is willing -or financially able- to purchase an average priced house given their income and valuations for the house. Therefore, I believe that, in order to create a perfect matching, there exists a market-clearing price that is lower than the one offered in the current market today.
In order to bridge the article to the theory learnt, we will propose a bipartite graph, where the left-hand side contains 3 individuals looking for a house, and on the right-hand side, 3 houses which are up for sale. In order to prove that there is a possibility of a market-clearing price and hence a perfect matching, I will be using the Dutch auction pricing mechanism. I will construct the preferred-seller graph using simplified valuation (income) and house prices as are the following:
Valuations
Individual a = (37 (relates to average home value 2021), 25, 20)
Individual b = (25, 17, 7(relates to median household income 2021))
Individual c = (7, 6, 12)
Prices
House x = 37
House y = 37
House z = 37
Here, we can see that nodes y and z are constricted as no individual is able to purchase them. As a result, I will lower their prices until another individual is able to purchase the house (b values y at 17, so prices for y and z will be lowered to 17).
However, now a also wants house y as how their payoff would be (25-17 = 8) instead of (37-37 = 0) for house x. Therefore, sets x and z are now constricted. In order to fix this, I will make the price of house x = 29, therefore the payoff of picking either house x or y will be the same for individual a.
Although this fixes constricted set x, set z is still constricted. In order to change this, I will decrease the price of house z to 12 (individual c values house z at 12).
Now we are able to see that, with these prices, there is a perfect matching, where individual a gets house x, individual b gets house y and individual c gets house z. This is also an optimal matching as the individual with the highest valuations (a) got the most expensive house (x), the individual who had the second-highest valuation (b) got the second most expensive house (y) and the individual with the lowest valuations got the cheapest house (z), creating maximum total valuation of assignment.
However, as stated earlier, price 37 is derived from the average home value in 2021 ($374,900). Yet, as we can see, in order to reach a perfect matching, none of the houses in this example went past the price of $29. Therefore proving that, in my hypothetical situation, with the current average housing prices to median household income ratio, we will have a maximum of 1 matching (a-x).
News article:
- Delgado, Michelle. “U.S. House Prices Are Rising Exponentially Faster than Income (2021 Data).” Real Estate Witch, 22 Oct. 2021, https://www.realestatewitch.com/house-price-to-income-ratio-2021/.