Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 20 November 13 2008 So far, we ve considered matching markets in settings where there is no money you can t necessarily pay someone to marry you instead of another guy, and you can t necessarily pay Harvard to admit you instead of another student. In some settings, though, it s natural to assume money is a factor if you re choosing between two jobs, you may have preferences over the jobs themselves (location, hours, work environment), but they may also be offering you very different salaries and they may increase their offers as they compete over you. So we turn now to many-to-one matching with money, which is generally seen as a model of firms hiring workers Crawford and Knoer generalize the deferred acceptance algorithm to a setting with firms, workers, and money, but under two strong assumptions: firms production is additive as a function of the workers it hires, and each firm has an exogenous size cap. Kelso and Crawford ( Job Matching, Coalition Formation, and Gross Substitutes ) eliminate these two assumptions, replacing them with a much less restrictive condition, and show that everything still works. That s the paper we ll look at today. m workers 1,..., m, and n firms 1,..., n Each worker i has preferences over employers and wage, and gets utility u i (j, s) from working for firm j at a salary s this is assumed to be strictly increasing and continuous in s (doesn t have to be separable) Each firm has productivity that s a function of the set of workers it hires: if firm j hires a set of workers C j and pays them each s ij, its payoff is y j (C j ) i C j s ij Kelso and Crawford make two innocuous assumptions and one significant one. First, they define σ ij as the lowest salary at which worker i would work for firm j as opposed to remaining unemployed, so u i (j, σ ij ) = u i (0, 0); they assume that for any j, any C and any i / C, y j (C j {i}) y j (C) σ ij (This seems strong, until they point out that a firm could just pay someone zero to sit at home and do nothing.) 1
Second, they assume a firm with no workers doesn t produce y j ( ) = 0 And most importantly, they require that each firm s production function satisfies the gross substitutes condition Formally, for a given vector of prices s j = (s 1j, s 2j,..., s mj ), let M j (s j ) solve the firm s profit maximization problem, that is, { M j (s j ) = arg max y j (C) } s ij C i C Let s j be another vector of prices, weakly higher than s j, that is, s j s j. And for any C j, let T j (C j ) = { i C j : s ij = s ij } Then gross substitutes requires that if C j M j (s j ), there exists C j M j ( s j ) such that T j (C j ) C j. This is the same substitutes condition we saw in multi-unit auctions. Start out at a set of prices s j, and choose your favorite set of workers. Now raise the prices of some workers, leaving others the same. Gross substitutes says that you still want the workers whose prices stayed the same. (The definition is a bit clunkier just because you might be indifferent among more than one possible set of workers it says there has to be an optimal choice that includes all the old workers whose prices didn t go up.) 2
Recall Tuesday, we showed the equivalence between the set of stable matchings in the college admissions model and the core defined by weak dominance Here, the same equivalence holds, so Kelso and Crawford talk only about the core Define an allocation as an assignment of workers to firms, f : {1,..., m} {1,..., n}, and a set of salaries s if(i) An allocation is individually rational if u i (f(i), s if(i) ) u i (0, 0) (each worker prefers his deal to unemployment), which is the same as s if(i) σ if(i) ; and no firm makes negative profits, y j (C j ) i C j s ij where C j = {i j = f(i)} Kelso and Crawford use funny language for distinguishing the regular core, defined by strict dominance, from the core defined by weak dominance they call the latter the strict core An allocation is in the strict core if it is individually rational and there is no firm j, and set of workers C, and set of salaries r j that satisfy for each i C and u i (j, r ij ) u i (f(i), s if(i) ) π j (C, r j ) π j (C j, s j ) with at least one of these inequalities holding strictly The allocation is in the core if there is no j, C such that all these equations hold strictly Now, when money is continuous, there is no distinction between the core and the strict core as long as one person is strictly better off, they could give away a little money and still be strictly better off, so you could give everyone else a little bit more money, so everyone ends up strictly better off However, the proofs in Kelso and Crawford involve making money discrete, in which case the two can be different An allocation is in the discrete strict core, or the discrete core, if there is no such deviating coalition when the salaries r are all required to be on the discrete grid (say, integers) (Also note that the reason they name them the way they do is that the strict core is smaller than the regular core, since it s easier for something to not be in the strict core) One other solution concept Kelso and Crawford mention is a competitive equilibrium. This is when there is a price (wage) s ij for every firm-worker pair, and at those prices, the workers choose their favorite firm, the firms choose their favorite sets of workers, and the market clears (each worker demands the firm that demands him). These can be thought of as allocations where every firm and worker who aren t matched together, still know the price at which they could match together, but they re both content not to. 3
On, then, to results Like in the other papers we ve seen recently, Kelso and Crawford prove the existence of a core allocation by giving an algorithm to find one But first, they make money discrete that is, transfers can only occur in multiples of some amount (say $1) For any discrete-money model, they give an algorithm to find a core allocation the algorithm is a generalization of the deferred-acceptance algorithm Then they show that if the core was empty when money was continuous, it would also have to be empty when money is discrete but small enough, so there you go So, suppose salaries must be denominated in whole dollars, and let s see how the algorithm works It s actually very similar to the simultaneous-ascending-auction that we saw earlier, except that each firm faces a different set of personalized prices In each round, each firm j has a set of permitted prices (s 1j (t), s 2j (t),..., s mj (t)) at which it s allowed to make a job offer to each worker These prices start out at s ij (0) = σ ij, and go up over the course of the algorithm In the first round, by assumption, firms would be willing to hire everyone at those prices, so they are required to make job offers to everyone In each later round, facing a set of permitted prices (s ij (t)), firm j chooses the set of workers that maximizes profits and makes offers to them. Indifferences can be resolved in any way, except that if a firm made an offer to a worker in the previous round and that worker s price to that firm did not go up, they have to make an offer to him again. (Gross substitutes is exactly the condition that there is at least one profit-maximizing set of workers at the new prices that contains all the old workers whose prices didn t rise.) In each round, workers who get one or more job offer reject all but their favorite one, and tentatively accept their favorite If, in round t, firm j made an offer to worker i at price s ij (t) and that offer was rejected, then that firm s permitted price goes up by one increment that is, s ij (t + 1) = s ij (t) + 1 If not if in round t, firm j didn t make an offer to worker i, or made an offer that was tentatively accepted, the price stays the same s ij (t + 1) = s ij (t) We keep going until a round in which no offers are rejected, then we stop and the offers in place are accepted 4
So this is basically the firm-proposing deferred acceptance algorithm, with the twist that proposals come with wage offers, and that the lowest wage a firm can offer a given worker goes up each time an offer to that worker is rejected And it s analogous to the simultaneous ascending auction model with straightforward bidding each bidder (firm) faces a vector of personalized prices, and chooses the package that maximizes their payoff at those prices Theorem 1 from Kelso and Crawford: This process (they call it the salary adjustment process) converges in finite time to an allocation in the discrete core of the discrete model First, note that every worker always has at least one offer, so there is no unemployment This is because every worker gets offers in the first round (at σ ij ), nobody ever rejects all their offers, and any unrejected offer is repeated the next period Second, the process stops in finite time This is because in every round where the process continues, at least one permitted price s ij must increase by 1. We can put an upper bound on the wage any firm would ever pay any worker; so in finite time, if the algorithm is still going, no firm would be willing to hire anyone Third, when the process stops, it s at an individually rational allocation No worker tentatively accepts an unacceptable offer; in the round when the algorithm ends, no offers are rejected, and no firm ever makes a set of offers it wouldn t stand by Finally, the big result: when the algorithm ends, it s at a discrete core allocation Let φ be the matching the process stopped at, and s 1φ(1), s 2φ(2),... be the wages Suppose this is not in the discrete core Since it s individually rational, that means there must be some firm j, group of workers C, and set of salaries r j which block it, that is, such that u i (j, r ij ) > u i ( φ(i), s iφ(i) (t ) ) and ( ) π j (C, r j ) > π j C j φ, sj (t ) where t is the round when the algorithm stopped Now, if u i (j, r ij ) is better than worker i s outcome under φ, then worker i must never have rejected an offer from firm j at wage r ij or higher Which means the highest wage worker i every rejected from firm j was at most r ij 1, so the permitted price for firm j at time t was s ij (t ) r ij 5
But this holds for every worker i in C If the set of workers in C, at wages r ij, would have given firm j strictly higher profits than he got in round t of the algorithm, well then, that s who he would have proposed to! So we re done Next, Kelso and Crawford prove that when money becomes continuous, the core is still nonempty The proof is by contradiction: basically, they show that if you have a continuous market with an empty core, you could construct a discrete market with a small enough unit of money and that would have to have an empty core too Take any individually rational allocation ( φ, s 1φ(1),..., s mφ(m) ) in a continuous market that is not in the core Let D(j, C) be the amount that firm j and workers C could collectively gain by deviating by assumption, there is some j and C such that D is strictly positive Define F as the max of all the D, as a function of the allocation ( φ, s 1φ(1),..., s mφ(m) ) (so F is the maximum that any coalition could gain by deviating from the allocation φ) So we know F is strictly positive everywhere F is continuous in the wages s iφ(i), and we can make the space of money compact (by bounding wages above by the max profits of any firm), so the min of F over all possible wages is strictly positive And there are only a finite number of possible matchings φ, so the min over those exists and is strictly positive So the min of F over all possible allocations is bounded away from 0 that is, there is some number ɛ such that for any allocation in this continuous market, there is some coalition that can gain at least ɛ collectively by deviating 1 So make the unit of money smaller than m+1ɛ, and you re guaranteed a deviating coalition in the discrete market too But we already proved discrete markets can t have empty cores, and we re done. Given a core allocation, finding a competitive equilibrium set off-equilibrium wages to make workers indifferent 6
For a given discrete grid, indifferences are nongeneric; if we assume them away, we re guaranteed a unique path on the salary adjustment process, which is nice In that case, the algorithm converges to a discrete core allocation that is at least as good for every firm as any other This is the extension of what we already knew: the hospitals-proposing algorithm converges to the hospital-optimal stable matching The proof is similar, we ll skip it They also point out that there can be only be a single firms-optimal core allocation This allows Kelso and Crawford to prove some cool comparative statics when it comes to adding firms or workers to the market Basically: if you add a firm, all the workers are weakly better off, and all the firms are weakly worse off And if you take away a worker, all the other workers are weakly better off, and all the firms are weakly worse off The way they prove these is pretty cool Suppose we first run the salary adjustment process with n firms, and reach the end point Now we drop in one more firm, firm n + 1; allow him to start making offers at s i,n+1 (t + 1) = σ i,n+1 ; and restart the algorithm Similar to the original proof, when the restarted algorithm ends, it must be at a firm-optimal discrete core allocation in the new market with n + 1 firms But since there is a unique firm-optimal discrete core allocation, this is the same one as if we had just started out with n + 1 firms And since restarting the algorithm could only make workers better off they start at time t + 1 with the old job offer they would have accepted before workers are better off after the firm is added As for firms, since permitted prices only rise over time, whatever allocation they end up with at the end now, they could have had at least as cheaply before firm n + 1 entered, so they re weakly worse off 7
As for removing a worker... Again, start off with n firms and m workers, and run the algorithm until it stops Now at time t + 1, change every firm s permitted salary for worker m to some huge number K And restart the algorithm Wherever it stops, nobody hires worker m And this is a firm-optimal discrete core allocation for the market with m 1 workers And, since every step of the restarted algorithm makes workers better off and firms worse off, workers 1 through m 1 are all weakly better off than in the market with m workers, and the firms are all weakly worse off WE ENDED HERE WILL CONTINUE TUESDAY WITH EXAMPLE Kelso and Crawford also explore a bit the gross substitutes condition When all the workers are equivalent that is, firms don t care who they hire, just how many gross substitutes is exactly equivalent to decreasing returns the condition that y j (w + 1) y j (w) y j (w) y j (w 1) When the workers are not equivalent but m = 2 there are just two workers gross substitutes is equivalent to the production functions being subadditive that is, y j ({1, 2}) y j ({1}) + y j ({2}) However, gross substitutes and subadditivity are not equivalent when there are more than two workers An example: suppose there are three workers, 1, 2, and 3. Firm j s production technology is y j ( ) = 0 y j ({1}) = 4 y j ({2}) = 4 y j ({3}) = 4 1 4 y j ({1, 2}) = 7 1 2 y j ({1, 3}) = 7 y j ({2, 3}) = 7 y j ({1, 2, 3}) = 9 Clearly subadditive the first worker is always worth at least 4, second worker is worth more than 2 and less than 4, third worker is worth 2 or less. 8
At prices (3, 3, 3), firm j demands workers 1 and 2 At prices (3, 4, 3), firm j demands worker 3 only So an increase in the price of worker 2 causes the firm to abandon worker 1 So gross substitutes is violated Kelso and Crawford use this firm to generate an example where gross substitutes is violated and the core is empty (there are no stable matchings) Add a second firm k with a similar production technology: y k ( ) = 0 y k ({1}) = 4 1 4 y k ({2}) = 4 y k ({3}) = 4 y k ({1, 2}) = 7 y k ({1, 3}) = 7 y k ({2, 3}) = 7 1 2 y j ({1, 2, 3}) = 9 Assume u i (j, s) = u i (k, s) = s each worker just cares about his salary If there is a core allocation, it must maximize total surplus, otherwise the coalition of both firms and all workers could deviate This means it must either assign 1 to firm j and 2 and 3 to k, or 1 and 2 to j and 3 to k Since these are symmetric, it s enough to show one of them is not in the core consider the first, firm j gets 1 and firm k gets 2 and 3 Roth and Sotomayor (chapter 6) show that gross substitutes is not enough when firms have budget constraints they give an example where gross substitutes is violated, firms are budget-constrained, and the core is empty Tuesday: Hatfield and Milgrom, and Ostrovsky 9