Market Equilibrium#
Previously, we studied situations with no transfers:
If Stanford housing auctioned rooms, the distribution would look much different.
It is illegal in all countries (except Iran) to buy organs.
Hospitals and students matching with money is subject to anti-trust laws.
However, there are issues to restricting markets to operating without money.
Can only work with ordinal opposed to cardinal preferences.
Underground markets may arise (college admissions scandals, black markets for kidneys).
Could be issues with permissions (entering donor lists multiple times using bots to try to get a match earlier).
We have covered four types of game equilbiria:
Nash
Correlated
Stackelberg
Subgame Perfect Equilibrium (SPE)
Today, we begin covering market equilibria.
Definition 19 (Cardinal And Ordinal Utility)
There are two main ways to model utility:
Cardinal: assigns some numeric value to how much someone likes something.
Ordinal: comparison-based, only assigns some order.
Cardinal utilities tell us more (we can always extract ordinal utilities from cardinal utilities, but not the other way around) but humans think more in terms of ordinal rankings. Cardinal utilities are also easier to work with when doing expected utility calculations and can be normalized to be in terms of (usually but not always) money.
Definition 20 (Fungible And Idiosyncratic Goods)
There are two types of goods:
Fungible goods have many interchangeable units for sale.
Idiosyncratic goods are unique.
Many goods are somewhere in between: rides on ride-sharing apps are fungible in terms of how the driver doesn’t matter too much, but can also be idiosyncratic in terms of caring about where you go.
Definition 21 (Supply And Demand Curves)
From economics,
A demand curve plots the total quantity of a good consumers are willing to buy at all price levels.
A supply curve plots the total quantity of a good firms are willing to sell at all price levels.
By tradition, price is plotted on the \(y\)-axis and quantity is plotted on the \(x\)-axis.
Remark 7
In general, demand is downward sloping (higher quantity demanded as price decreases) and supply is upwards sloping (lower quantity supplied as price decreases). However, this is not always the case (different market structures like a monopoly, weird classes of goods).
Definition 22 (Market Clearing Price)
The price where the supply and demand curves meet is called the market clearing price. At this price, the quantity demanded is equal to quantity demanded so all interested buyers and sellers can transact (and hence clear the market).
Unit-Demand Market Model#
Setting:
\(m\) different (idiosyncratic) goods for sale;
\(n\) different buyers that will each purchase at most one good (they have unit demand);
Buyer \(i\) as value \(v_{i,j}\) for purchasing good \(j\);
If buyer \(i\) pays \(p_j\) to acquire good \(j\), then their payoff is
\[U_{i,j} = v_{i,j} - p_j;\]If buyer \(i\) doesn’t purchase any good, normalize outside utility to be zero:
\[U_{i, \varnothing} = 0.\]
The solution concept we will use:
Definition 23 (Competitive Equilibrium)
A competitive equilibrium is a price vector \(p = (p_1,...,p_m)\) and a matching \(M: \{1,2,...,n\} \to \{1,2,...,m\}\) of buyers to goods such that:
Each buyer is matched with their favorite good given prices \(p\): for all \(i,j\) we have
\[v_{i,M(i)} - p_{M(i)} > v_{i,j} - p_j;\]If no buyer is matched with good \(j\) then \(p_j = 0\);
Buyer \(i\) is unmatched if for all goods \(j\), we have \(v_{i,j} - p_j < 0\).
Remark 8
Conditions (1) and (3) implies that buyer \(i\) is unmatched if and only if for all goods \(j\), we have \(v_{i,j} - p_j < 0\). Then, this stronger condition is equivalent to individual rationality: for all buyers \(i\),
In addition to equilibrium, we also care about welfare:
Remark 9
Prices are not included because the total cost to buyers is equal to total profit of sellers.
Properties of Competitive Equilibrium#
Theorem 18
If \((p,M)\) is a competitive equilibrium, then \(M\) is a matching that maximizes social welfare: if \(M'\) is any matching, we have
Proof. By the definition of competitive equilibrium, we know
as \(v_{i,M(i)}-p_{M(i)} \geq v_{i,M'(i)}-p_{M'(i)}\) for all \(i\). Then, \(\sum_i p_{M(i)} = \sum_i p_{M'(i)}\) as \(M,M'\) are just permutations so we get
Theorem 19
In the unit-demand market model with finite prices and finite increments, there will always exist a competitive equilibrium.
Remark 10
In more nuanced models, competitive equilibrium does not necessarily exist.
Proof. We will construct a Deferred Acceptance with Prices algorithm that gets us to competitive equilibrium. For each buyer, construct a list of all (good, price) options ordered by utility (we can truncate this list by not including anything worse than (receive nothing, pay nothing), which is always an option). At each iteration of the algorithm, every unmatched buyer whose next-favorite option is \((j,p)\) proposes price \(p\) to good \(j\). Then, good \(j\) tentatively accepts \(p\) (in the deferred acceptance sense) if it is higher than the prices it was previously offered. This algorithm will always terminate by the same reasoning as why deferred acceptance always terminates. The resulting match and prices when this algorithm terminates is a competitive equilibrium:
Every buyer is matched with their favorite (good, price) pair as any other pair must have led to the buyer being rejected at some previous stage.
Each unmatched good has a price of zero as a good is unmatched if and only if it was never proposed to.
Remark 11
DA with prices runs in \(O(n \cdot m \cdot k)\) time where \(k\) is the number of price increments.
Proposition 1
Deferred acceptance with prices is strategyproof and optimal for buyers.
Proof. Follows directly from the same reasoning used to analyze deferred acceptance.
Definition 24 (Social Welfare)
The social welfare of an allocation \(M\) is the sum of buyers’ values: