EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 - - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 5

Leonardo Felli

CLM.G.4

6 December 2011

Property Rights Theory

The presence of inefficiencies due to the incompleteness of contracts naturally leads to the question: Do there exist institutions that ameliorates the inefficiencies that are generated by contractual incompleteness? The answer we are going to give is a positive one, in particular one of these institutions is the building block of modern economy: ownership

• f physical assets.

The analysis of this institutions will also help us to answer a key question for economic analysis: What is a firm? What determines the boundaries of a firm?

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 2 / 50

Property Rights Theory (2)

We will see that while ownership rights are irrelevant when the Coase Theorem holds this is no longer the case when frictions prevent the Coase Theorem from holding at least at an ex-ante stage. The theory we present is known as property rights theory (Grossman and Hart 1986, Hart and Moore 1990). Ingredients:

incomplete contracts, and hence the presence of a potential hold-up problem, relationship specific investments, the lack of a well-functioning market for the good that embodies the parties’ investments.

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Specific Investments

Question: why specificity plays a critical role. To understand this consider the following simple hold-up problem. A buyer and a seller engage in trade. At an ex-ante stage the seller can undertake an ex-ante investment e that enhances the quality of the unit of good to the buyer at a cost e2 2 The value to the buyer is then v e Assume for simplicity that the delivery cost for the seller c = 0.

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Incomplete Contracts

We assume an extreme incomplete contract framework: no contract will be drawn before the seller chooses the investment. Timing:

The seller undertakes the ex-ante investment e; A contract is agreed upon between the buyer and the seller to trade at a given price p; Trade occurs.

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Efficient Outcome

Ex-post efficiency requires that trade occurs as long as e ≥ 0 from v e ≥ 0. Ex-ante efficiency requires that e∗ is the solution to the following problem: max

e

v e − e2 2 ,

• r

e∗ = v Assume first that at the contracting stage the buyer makes a take-it-or-leave-it offer to the seller. We solve backward for the SPE of this simple game.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 6 / 50

Underinvestment

Let e be given, at the contracting stage the gains from trade are: v e. The seller will accept the offer if and only if: p ≥ 0 The buyer’s offer is then obviously ¯ p = 0 The seller’s ex-ante investment is then such that: max

e

¯ p − e2 2 = −e2 2 In other words, the hold-up problem generated by incomplete contracts implies: ¯ e = 0 < e∗ = v .

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 7 / 50

Underinvestment (2)

In general assume that the parties’ bargaining power is such that the seller gets λ of the gain from trade while the buyer gets (1 − λ) of the gains from trade: λ(v e), (1 − λ)(v e) The seller’s investment problem is then: max

e

λ(v e) − e2 2 In other words we obtain that (hold-up problem): e∗∗ = λ v ≤ e∗ = v In particular, the equality will hold only if the seller has full bargaining power.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 8 / 50

Market for the Good

Assume now that instead of selling to a unique buyer in a situation of bilateral monopolism, there is a market for the good. In particular, assume that there are two potential buyers for the good. Both buyers value the good v e and they Bertrand compete for the good. Let bi be the bid of buyer i ∈ {1, 2} for the good.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 9 / 50

Bertrand Competition

Extensive form of the Bertrand competition game: the good is allocated to the buyer who makes the highest bid, if the bids are equal buyer 1 gets the good; the buyer who gets the good pays his bid to the seller. Buyer i’s expected payoff is then: πi =        v e − bi if bi > b−i and if bi = b−i and i gets the good if bi < b−i and if bi = b−i and i does not get the good

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 10 / 50

Bertrand Competition (2)

Buyer i’s best reply is then to choose:            v e > bi > b−i if b−i < v e and when bi = b−i then i does not get the good bi = b−i if b−i ≤ v e and when bi = b−i then i gets the good bi < b−i if b−i ≥ v e All Nash equilibria of this Bertrand competition game are such that: bi = b−i = v e

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 11 / 50

General Investment

Then the unique equilibrium price paid to the seller is: ˜ p = v e The seller’s ex-ante investment problem is then: max

e

˜ p − e2 2 = v e − e2 2 The first order conditions imply: ˜ e = v = e∗ In other words, even in the presence of incomplete contracts the existence of a market for the commodity implies that the investment is efficient: ex-ante efficiency arises.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 12 / 50

Back to Specific Investments

Assume now that the seller’s investment is specific to the buyer. The investment e has different returns depending on whether the buyer is 1 or 2: v ∈ {v1, v2}, v1 > v2 If buyer 1 buys the good then the returns are v1 e while if buyer 2 buys the good the returns are v2 e. Assume the same Bertrand competition game.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 13 / 50

Bertrand Competition Again

Buyer i’s expected payoff is then: πi =        vi e − bi if bi > b−i and if bi = b−i and i gets the good if bi < b−i and if bi = b−i and i does not get the good Buyer i’s best reply is then to choose:            v e > bi > b−i if b−i < vi e and when bi = b−i then i does not get the good bi = b−i if b−i ≤ vi e and when bi = b−i then i gets the good bi < b−i if b−i ≥ vi e

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 14 / 50

Equilibrium of Bertrand Competition

Then the unique (trembling-hand-perfect, cautious) Nash equilibria of this Bertrand competition game is such that:

the buyers’ equilibrium bids are such that: b1 = b2 = v2 e buyer 1 gets the good.

Notice that the last condition implies ex-post efficiency.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 15 / 50

Underinvestment with a Market

Notice that in this case the equilibrium price paid to the seller is such that: ˆ p = v2 e The seller’s investment problem is then: max

e

ˆ p − e2 2 = v2 e − e2 2 The first order conditions of this problem imply: ˆ e = v2 < e∗ = v1 In other words, specificity of the ex-ante investment implies a hold-up problem and hence ex-ante inefficiencies even in the presence of a market for the good.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 16 / 50

Competition

Objective: to analyze the role that market competition for a match has in determining the parties’ outside option in the bargaining stage. Question: can this market competition reduce the inefficiency of the hold-up problem? Answer: yes but possibly at the cost of a related inefficiency. Coordination failure: a situation in which a group of agents can realise a mutual gain only by a change in behaviour by each member of the group. Ingredients: complementarities among parties’ investments and incomplete contracts.

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Ingredients

Observation: given the specific nature of investments where is the competition coming from? Competition: Bertrand competition among heterogeneous agents. We assume that workers bid for firms. Specificity: we assume a discrete number of heterogeneous workers that matches with a discrete number of heterogeneous firms. Complementarity: Both workers and firms undertake complementary investments. Contracts: we take contracts to be incomplete, they are bilateral agreements to trade at a constant price (wage).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 18 / 50

Results

Workers undertake constrained (by the match) efficient investments. Coordination failures may arise that take the form of additional equilibria (besides the efficient one) with inefficient matches. Firms undertake constrained inefficient investments. These investment are near-efficient. No coordination failures arise: the unique equilibrium exhibits constrained efficient matches (trade-off).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 19 / 50

A Matching Model

S workers s = 1, . . . , S match with T firms t = 1, . . . , T: S > T; s indexes workers innate ability: s = 1 highest; t indexes firms innate ability: t = 1 highest; both workers and firms choose match-specific investments: xs and yt Investemnt have an increasing and convex cost C(·);

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 20 / 50

Complementarities

workers’ and firms’ qualities: σ(s, xs) τ(t, yt); decreasing in s and t and increasing and concave in xs and yt. single crossing conditions: σ12 < 0, τ12 < 0. the surplus of each match v(σ, τ) increasing and concave in σ and τ. complementarity: v12 > 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 21 / 50

Period 1: Investment Game

workers choose investments xs simultaneously and independently; firms choose investments yt simultaneously and independently; firms and workers decision are also taken simultaneously and independently; the game moves to the second period.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 22 / 50

Period 2: Bertrand Competition Game

All workers simultaneously and independently make wage offers to every one of the T firms. Firm t = 1 (the most efficient) observes the offers she receives and decides which offer to accept. This commits the worker selected to work for firm 1 and automatically withdraws all offers this worker made to other firms. All other firms and workers observe this decision and then firm t = 2 decides which offer to accept. This process is repeated until firm T decides which offer to accept (generalized).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 23 / 50

Bertrand Competition Game

We look for the trembling-hand-perfect (cautious) equilibrium of the game. Let the ordered vectors of qualities be given: (τ1, . . . , τT), where τ1 > . . . > τT, (σ1, . . . , σT), where σ1 > . . . > σT.

Result

Every equilibrium of the Bertrand competition game is characterized by assortative matching: (σk, τk), ∀k = 1, . . . , T.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 24 / 50

Bertrand Competition Game

Proof: By contradiction. Equilibrium matches: (σ′, τ ′′),(σ′′, τ ′) where σ′ < σ′′ and τ ′ < τ ′′. Let B(τ ′) and B(τ ′′) be equilibrium accepted bids by firm τ ′ & τ ′′. Necessary conditions for (σ′′, τ ′) and (σ′, τ ′′) to be equilibrium matches: v(σ′′, τ ′) − B(τ ′) ≥ v(σ′′, τ ′′) − B(τ ′′); v(σ′, τ ′′) − B(τ ′′) ≥ v(σ′, τ ′) − B(τ ′). Both inequalities imply: v(σ′′, τ ′) + v(σ′, τ ′′) ≥ v(σ′′, τ ′′) + v(σ′, τ ′). A contradiction to the complementarity assumption.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 25 / 50

Runner-up

Definition

The runner-up worker to the firm τt is the worker σr(t) such that σr(t) ∈ {σt+1, . . . , σS} and has the highest willingness to pay for firm τt.

Result

The runner-up worker to firm τt is such that: σr(t) = max {σi | i is un-matched, σi ≤ σt} . The equilibrium payoffs to each firm and worker, for t = 1, . . . , T, are: πW

σt

= [v(σt, τt) − v(σr(t), τt)] + πW

σr(t)

πF

τt

= v(σr(t), τt) − πW

σr(t)

and for every i = T + 1, . . . , S: πW

σi = 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 26 / 50

Runner-up (2)

Proof: By induction. Consider firm τt. Worker σt’s maximum willingness to bid is v(σt, τt) − ˆ πW

σt ;

where ˆ πW

σt = v(σt, τr(t)) − v(σr2(t), τr(t)) + πW σr2(t);

• r

v(σt, τt) − v(σt, τr(t)) + v(σr2(t), τr(t)) − πW

σr2(t).

Worker σr(t)’s maximum willingness to bid is v(σr(t), τt) − πW

σr(t).

where πW

σr(t) = v(σr(t), τr(t)) − v(σr2(t), τr(t)) + πW σr2(t);

• r

v(σr(t), τt) − v(σr(t), τr(t)) + v(σr2(t), τr(t)) − πW

σr2(t).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 27 / 50

Runner-up (3)

Therefore v(σt, τt) − ˆ πW

σt > v(σr(t), τt) − πW σr(t)

since, from v12 > 0: v(σt, τt) − v(σt, τr(t)) > v(σr(t), τt) − v(σr(t), τr(t)). We get: Bσt = Bσr(t) = v(σr(t), τt) − πW

σr(t)

while payoffs are πW

σt

= [v(σt, τt) − v(σr(t), τt)] + πW

σr(t)

πF

τt

= v(σr(t), τt) − πW

σr(t).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 28 / 50

Equilibrium of Bertrand Competition Game

Result

Given the ordered vectors (σ1, . . . , σS), (τ1, . . . , τT), if firms select bids in the order of their qualities the unique equilibrium of the Bertrand competition game is such that the runner-up worker to firm τt is: σr(t) = σt+1 πW

σt

=

T

• h=t

[v(σh, τh) − v(σh+1, τh)] πF

τt

= v(σt+1, τt) −

T

• h=t+1

[v(σh, τh) − v(σh+1, τh)]

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 29 / 50

Observations:

The order in which firms select their bid is determined by an endogenous variable. Worker σt’s equilibrium payoff is: πW

σt = v(σt, τt) + Wσt

where Wσt is independent of σt. Firm τt’s equilibrium payoff is πF

τt = v(σt+1, τt) + Pτt.

where Pτt is independent of τt. In the general case (randomly determined order) the difference is in the identity of the runner-up worker.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 30 / 50

Properties

Result

Given the ordered vectors of investments (σ1, . . . , σi−1, σi+1, . . . , σS), (τ1, . . . , τi−1, τi+1, . . . , τS) the net payoffs to worker i below is continuous in σ: πW

i (σ)

= v(σ, τi) − v(σi+1, τi) + +

T

• h=i+1

[v(σh, τh) − v(σh+1, τh)] − C(x(i, σ)) and the net payoff to firm i below is continuous in τ: πF

i (τ)

= v(σi+1, τ) − [v(σi+1, τi+1) − v(σi+2, τi+1)] − −

T

• h=i+2
• v(σ(h), τh) − v(σh+1, τh)
• − C(y(i, τ))

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 31 / 50

Workers’ Investment Game

Assume that σ(s, xs) and τt = τ(t). Clearly the order of firms’ innate abilities coincides with the order of their qualities, hence the Result above applies. Given a match (s, t) the workers’ investment choice xs(t) is defined as: xs(t) = arg max

x

πW

σ(s,x) − C(x)

= arg max

x

v(σ(s, x), τt) − Wσ(s,x) − C(x).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 32 / 50

Workers’ Investment Game (2)

Result

Worker s’s investment choice xs(t) is constrained efficient (given the match (s, t)). Proof: The worker is residual claimant of the match surplus. Notice that from the definition of xs(t) we obtain also: d σ(s, xs(t)) d s < 0, d σ(s, xs(t)) d t < 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 33 / 50

Workers’ Investment Game (3)

Definition

A permutation (s1, . . . , sS) of (1, . . . , S) is an equilibrium of the workers’ investment game if and only if: σ(i) = σ(si, xsi(i)) < σ(i−1) = σ(si−1, xsi−1(i − 1)). where (σ(1), . . . , σ(S)) is an ordered vector.

Result

The equilibrium with efficient matches of the workers’ investment game characterized by si = i, i = 1, . . . , S always exists.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 34 / 50

Workers’ Investment Game (4)

Proof: The net surplus to each worker πW

i (σ) − C(x(i, σ))

is continuous in σ at σ(j), j = 1, . . . , S; is strictly concave with a local maximum at σ(i) in the interval (σ(i+1), σ(i−1)); is monotonic increasing in any interval (σ(k), σ(k−1)), for ever k = i + 2, . . . , S, to the left of (σ(i+1), σ(i−1)); is monotonic decreasing in any interval (σ(h), σ(h−1)), for ever h = 2, . . . , i − 1, to the right of (σ(i+1), σ(i−1)).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 35 / 50

Workers’ Investment Game (5)

Notice that in this equilibrium there are no hold-up problems and no coordination failures.

Result

Given (τ1, . . . , τT), it is possible to construct an inefficient equilibrium of the workers’ investment game such that si < si−1.

Result

Given (σ(s1, ·), . . . , σ(sS, ·)), it is possible to construct (τ1, . . . , τT) such that there does not exist any inefficient equilibrium of the workers’ investment game.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 36 / 50

Workers’ Investment Game (6)

Proof: If two workers have very similar innate abilities the difference in workers’ qualities is determined mainly by the difference in firms’ qualities. Then an argument similar to the one of the Result above applies and an inefficient equilibrium exists. If firms have very similar innate abilities then the equilibrium condition for an inefficient equilibrium is necessarily violated.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 37 / 50

Observations

When workers invest competition solves the hold-up problem. Inefficiencies may arise that take the form of coordination failures: equilibria with inefficient matches. This inefficiency is less likely the higher is the degree of specificity due to the workers’ characteristics with respect to the specificity due to the firms’ characteristics.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 38 / 50

Firms’ Investment Game

Assume that σs = σ(s) and τ(t, yt). Notice that in general in this case the Result above does not provide the characterization of the payoffs in the Bertrand competition game.

Result

If firms select their most preferred bid in the order of their innate abilities the unique equilibrium of the firms’ investment game is such that firm t chooses the simple investment: y(t, t + 1) = arg max

y

v(σt+1, τ(t, y)) − Pτ(t,y) − C(y). Proof: Notice first that the last firm to select its bid always chooses simple

• investments. Let t + 1 be the last firm to choose simple investments.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 39 / 50

Firms’ Investment Game (2)

Using the Result above we can identify the qualities (τt+1, . . . , τT) and payoffs in the t + 1 subgames of firms t + 1, . . . , T. σi τi τ ′

i

t σ1 τ1 τ ′

1

1 . . . . . . . . . . . . σt−2 τt−2 τ ∗

t

t − 2 σt−1 τt−1 τt−2 t − 1 σt τt τt−1 t σt+1 τt+1 τt+1 t + 1 . . . . . . . . . . . . σT τT τT T

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 40 / 50

Firms’ Investment Game (3)

Hence τk > τ ∗ > τt+1 for all k < t, implying that this Result applies and hence: y(t, t + 1) = arg max

y

v(σt+1, τ(t, y)) − Pτ(t,y) − C(y). Notice this Result implies that firm under-invest: y(t, t + 1) < y(t, t). There exist inefficiencies generated by hold-up problems. This Result also implies that the unique equilibrium of the firms’ investment game is characterized by efficient matches. In other words, there are no coordination failures.

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The Near-Efficiency of Firms’ Investments

Extra assumptions: responsive complementarity: Define y(t, s) = arg max

y

v(σs, τ(t, y)) − C(y) then ∂ ∂t ∂y ∂s

• > 0.

marginal complementarity: ∂2v2(σ, τ) ∂σ ∂τ > 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 42 / 50

The Near-Efficiency of Firms’ Investments (2)

Define now for convenience: ω(s, t) = v(σ(t), τ(t, y(t, s))) − C(y(s, t)). A measure of firm t’s induced inefficiency is: ω(t, t) − ω(t, t + 1). The overall inefficiency induced by firms’ under-investments is then: L =

T

• t=1

ω(t, t) −

T

• t=1

ω(t, t + 1). How large is the loss L?

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 43 / 50

The Near-Efficiency of Firms’ Investments (3)

Define M the efficiency loss resulting from firm 1 choosing an investment level when matched with worker 1. Firm 1’s investment choice is then: y(1, T + 1) Define now: M = ω(1, 1) − ω(1, T + 1).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 44 / 50

The Near-Efficiency of Firms’ Investments (4)

Result

Assume T ≥ 2. If both responsive complementarity and marginal complementarity hold then L < M. Proof: Firms have an incentive to invest to improve their outside option: ω(t, t + 1) + Pτt. The under-investment of each firm is relatively small and total inefficiency is then obtained by aggregating these relatively small under-investments. The decreasing returns to investment and the assumptions on how optimal firms’ investments change across different matches guarantees that L < M.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 45 / 50

Ex-Ante Competition

Properties of the firms’ investments depend on firms selecting their most preferred bid in the order of their innate abilities. Question: does this order arise endogenously if firms compete ex-ante for the order in which they select their most preferred bid? Answer: there always exists an equilibrium of the ex-ante competition among firms that leads to the firms selecting their most preferred bid in the order of their innate ability. Restrict attention to the equilibrium of the firms’ investment game (always exists) such that firms’ investments induce an order of firms’ qualities that coincides with the order of their innate abilities.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 46 / 50

Ex-Ante Competition (2)

Assume that there exists an equilibrium of the ex-ante competition stage such that the order in which firms select their bids is: τt∗ < τt∗+1 > τt∗+2 > . . . > τT Let t′ be such that t′ ∈ {t∗, t∗ + 1, . . . , T} and τt∗+1 > τt∗+2 > . . . > τt′ > τt∗ > τt′+1 > . . . > τT If firm τt∗ swaps place with firm τt′ then both firms gain. The runner-up worker for firm τt∗ it is still the worker σt′+1 that in equilibrium matches with firm τt′+1, hence the payoff of firm τt∗ does not change: πF

t∗ = ˜

πF

t∗.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 47 / 50

Ex-Ante Competition (3)

By one of the previous Results the runner-up worker to firm τt′ is now the worker σt∗ that in equilibrium matches with firm τt∗ (while before the change it was worker σt′+1). The difference in firm τt′ payoffs after (˜ πF

t′) and before (πF t′) the

change is then: ˜ πF

t′ − πF t′ = v(σt∗, ˜

τt′) − v(σt∗, ˜ τt∗) + v(σt′+1, ˜ τt∗) − v(σt′+1, τt′) By v12 > 0, ˜ τt′ > ˜ τt∗, ˜ τt′ > τt′ and σt∗ > σt′+1 we get: ˜ πF

t′ − πF t′ > 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 48 / 50

Ex-Ante Competition (4)

Result

If firms select their preferred bid in any order other than the decreasing

• rder of their innate abilities then there exists a pair of firms who gain —
• ne strictly and one weakly – from swapping positions in the order in

which they select their bid. The equilibrium in which firms select their bid in the order of their innate abilities is unique if firms’ ability are sufficiently far apart.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 5 6 December 2011 49 / 50

Ex-Ante Competition (5)

Bertrand competition between workers and firms for matches may help solve the hold-up problems at the cost of coordination failures. Notice that when both workers and firms undertake ex-ante investments:

workers’ investments are still constrained efficient; firms’ investments are still near-efficient.

However, inefficiency arise that takes the form of coordination failures. In other words, competition does a fairly good job at solving hold-up problems.

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