<p>If you want to implement investment levels that leave both better off than before, and if the output is contractible, then you can write a contract with a third party that will receive the whole output unless it reaches at least the prescribed level.</p>
<p>Say, you want to induce <span class="math-container">$i_1^*$</span> and <span class="math-container">$i_2^*$</span>, and both are better off investing this much than not participating. Assume also that at these investment levels, the individual marginal benefit is smaller than the marginal cost.</p>
<p>Now, if the output is at least <span class="math-container">$f(i_1^*,i_2^*)$</span>, the whole output is transferred to a third party, who will gladly accept it. They cannot lose. Then none of these two firms has an incentive to produce more or less than the prescribed investment level.</p>
<p>The argument relies of course on production being deterministic and output being fully contractible. I do not claim that this is always the best approach &quot;in reality.&quot;</p>

CC BY-SA 4.0
2022-05-22T08:55:13.933Z
2022-05-22T08:55:13.933Z
Michael Greinecker
<p>I add an answer to complement Michael Greinecker's. Unlike his answer, I consider a renegotiation design problem, whereby the two parties sign a contract prior to making investments which determines the ex-post bargaining over surplus. Hence, I endogenise the <span class="math-container">$\beta_1$</span> term you describe. This sort of contract is perhaps more realistic compared to one requiring large negative penalties for potential underinvestment.</p>
<p>This renegotiation design problem is effectively the same setup as Agion-Dewatripont-Rey (1994 ; Econometrica). They show that although one cannot make both parties residual claimant, one can chose one party to be full residual claimant whilst simultaneously making the other <span class="math-container">$\textit{locally}$</span> residual claimant. These contracts are known in the literature as specific performance contracts.</p>
<p>A simplified version of their model is as follows. Suppose there is a buyer and a seller. The buyer and seller trade over a single indivisible good, which is valued by the buyer at <span class="math-container">$v$</span> and which costs the seller <span class="math-container">$c$</span> to provide. At the time of trading, <span class="math-container">$v$</span> and <span class="math-container">$c$</span> are observable to both parties and we allow them to bargain over the terms of trade so that trade always occurs efficiently given <span class="math-container">$v$</span> and <span class="math-container">$c$</span>.</p>
<p>Prior to the realisation of <span class="math-container">$v$</span> and <span class="math-container">$c$</span>, the buyer and seller make ex-ante investments which affect the distribution of <span class="math-container">$v$</span> and <span class="math-container">$c$</span> (and so affect the return function). Suppose there are two types, which are ordered <span class="math-container">$c_H&gt;v_H&gt;c_L&gt;v_L$</span>. Hence, the buyer and seller only trade when the realisation is <span class="math-container">$(v_H,c_L)$</span>. The buyer's investment, <span class="math-container">$i$</span>, costs <span class="math-container">$\psi(i)$</span>, whilst the supplier's investment, <span class="math-container">$j$</span>, costs <span class="math-container">$\phi(j)$</span>, where <span class="math-container">$\psi,\phi$</span> are convex and increasing. These investments affect the distribution of types linearly: <span class="math-container">$\mathbb{P}(v=v_H)=i$</span>, <span class="math-container">$\mathbb{P}(c=c_L)=j$</span>.</p>
<p>The first-best level of investments maximises total welfare. <span class="math-container">$$(i^{FB},j^{FB})=\arg \max_{i,j}\{ij(v_H-c_L)-\psi(i)-\phi(j)\}$$</span> So, the first best investment levels are implicitly defined by <span class="math-container">$$\begin{align} i^{FB}(v_H-c_L)&amp;=\phi'(j^{FB}) \\
j^{FB}(v_H-c_L)&amp;=\psi'(i^{FB})\end{align}$$</span> Absent any other contracts, there will be Williamsonian underinvestment since each party is only partially residual claimant on the ex-post surplus.</p>
<p>What ADR do is define a renegotiation proof contract which determines the terms of renegotiation once <span class="math-container">$v$</span> and <span class="math-container">$c$</span> are realised. This contract is signed prior to the ex-ante investment stage. The idea of the contract is to allocate all bargaining power to one party (WLOG, the buyer) whilst giving the other party the option to request a default option - specified in the contract - which that party can request regardless of whether or not it is efficient to do so. This sort of contract makes the buyer full residual claimant whilst also making the seller locally residual claimant.</p>
<p>Suppose the default option, as specified in the contract, is a mechanism <span class="math-container">$(\tilde{q},\tilde{p})$</span>. After <span class="math-container">$(v,c)$</span> is realised, if the buyer is allocated bargaining power, it solves <span class="math-container">$$\max_{p,q}\{qv-p\} \qquad st. \qquad  p-qc\geq \tilde{p}-\tilde{q}c$$</span> The constraint binds at the optimum, so the problem reduces to <span class="math-container">$$\max_{q}\{q(v-c)-\tilde{p}+\tilde{q}c\}$$</span> Hence, the buyer chooses <span class="math-container">$\tilde{q}=\mathbb{1}_{\{v\geq c\}}$</span> so trade occurs efficiently. The buyer selects <span class="math-container">$p$</span> to hold the supplier down to its default option utility.</p>
<p>The supplier, knowing they will be held down to their default option utility, chooses their investment to solve <span class="math-container">$$\max_j\{\tilde{p}-\tilde{q}(jc_L+(1-j)c_H)-\phi(j)\}$$</span> So, the supplier picks investment level <span class="math-container">$j^*$</span> such that <span class="math-container">$$\phi'(j^*)=\tilde{q}(c_H-c_L)$$</span></p>
<p>Now, <span class="math-container">$\tilde{q}$</span> is free for us to choose. In particular, picking <span class="math-container">$\tilde{q}$</span> such that <span class="math-container">$\phi'(j^{FB})=\tilde{q}(c_H-c_L)$</span> means that the supplier picks the first best level of investment! What's more is that given the supplier picks the first-best investment level, the buyer maximises the total surplus minus the supplier's surplus, solving <span class="math-container">$$\max_{i}\{ij^{FB}(v_H-c_L)-\psi(i)-[\tilde{p}-\tilde{q}(j^{FB}c_L+(1-j^{FB})c_H)-\phi(j^{FB})]\}$$</span> Therefore, the buyer picks <span class="math-container">$$\psi(i^*)=j^{FB}(v_H-c_L)$$</span> and so <span class="math-container">$i^*=i^{FB}$</span>. Hence, there is no problem if we allow for a sufficiently rich renegotiation design. Noldeke and Schmidt (1995; RAND) show that we can get the same result if we only consider &quot;option contracts&quot; which allow the supplier to choose whether or not to provide the good at a prespecified price but allows the buyer to propose new price offers. The problem with these two approaches is that they require that the supplier is able to request inefficient outcomes. Whilst this never happens in equilibrium, it is still unappealing that it is possible. ADR give some justification that their contract can be considered as a Rubinstein bargaining game where the parties can at each step request a default option, which is a more appealing implementation.</p>
<p>However, we really do need the renegotiation contract space to be sufficiently rich. Hart and Moore (1988 ; Econometrica) show that if specific performance contracts are not possible the only default option that a party can request is &quot;no trade&quot; (also known as &quot;at-will&quot; contracting). In this case, the first best is generally not achievable and there will be underinvestment as a result.</p>

CC BY-SA 4.0
2024-07-06T09:29:20.020Z
2024-07-06T09:29:20.020Z
Joseph Basford
<p>Say two agents jointly invest with a return function <span class="math-container">$f(i_1, i_2)$</span>, which is increasing in both and concave, and they share the return by ratio <span class="math-container">$\beta_1$</span> and <span class="math-container">$1-\beta_1$</span>.</p>
<p>Then the result of investment level will be the solution of <span class="math-container">$\beta_1 f_1(i_1, i_2^*)=mc_1$</span> and <span class="math-container">$(1-\beta_1) f_2(i_1^*, i_2)=mc_2$</span>. And these investments will be inefficient: if all agents increase their investment by a little, it will make both better off.</p>
<p>This is different from the one-side hold-up problems because we can never given the full claimant on residual to both side to solve the incentive problem, and I somehow feel this is like a continuous version of  the Prisoner's Dilemma (Is there a formal name?). I think this general form is very common in the real world. I wonder do we have ways to resolve this holdup problem other than putting it into an infinite repeated horizon?</p>

CC BY-SA 4.0
2022-05-25T10:02:28.473Z
Maybe I am missing something but I don’t understand why this would be common in the real world. Shouldn’t the share depend on investment?
LudwigNagasena
CC BY-SA 4.0
2022-05-25T15:50:05.627Z
@LudwigNagasena For example you and your colleague jointly do a project in the firm. It is not easy for your supervisor to detect individual effort and the share is often decided by some pre-determined rule.
Alalalalaki
<game-theory>
Resolve Hold-Up Problems in Joint Investment
Alalalalaki
# Resolve Hold-Up Problems in Joint Investment

(Asked by: username_0 on 2022-05-21)

Say two agents jointly invest with a return function $f(i\_1, i\_2)$, which is increasing in both and concave, and they share the return by ratio $\beta\_1$ and $1-\beta\_1$.

Then the result of investment level will be the solution of $\beta\_1 f\_1(i\_1, i\_2^\*)=mc\_1$ and $(1-\beta\_1) f\_2(i\_1^\*, i\_2)=mc\_2$. And these investments will be inefficient: if all agents increase their investment by a little, it will make both better off.

This is different from the one-side hold-up problems because we can never given the full claimant on residual to both side to solve the incentive problem, and I somehow feel this is like a continuous version of the Prisoner's Dilemma (Is there a formal name?). I think this general form is very common in the real world. I wonder do we have ways to resolve this holdup problem other than putting it into an infinite repeated horizon?

**Tags:** game-theory
> **Comments:**
> **username_1**: Maybe I am missing something but I don’t understand why this would be common in the real world. Shouldn’t the share depend on investment?  
> **username_0**: @LudwigNagasena For example you and your colleague jointly do a project in the firm. It is not easy for your supervisor to detect individual effort and the share is often decided by some pre-determined rule.  

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## 2 answers

### Answer by username_2 on 2022-05-22. Score: 3

If you want to implement investment levels that leave both better off than before, and if the output is contractible, then you can write a contract with a third party that will receive the whole output unless it reaches at least the prescribed level.

Say, you want to induce $i\_1^\*$ and $i\_2^\*$, and both are better off investing this much than not participating. Assume also that at these investment levels, the individual marginal benefit is smaller than the marginal cost.

Now, if the output is at least $f(i\_1^\*,i\_2^\*)$, the whole output is transferred to a third party, who will gladly accept it. They cannot lose. Then none of these two firms has an incentive to produce more or less than the prescribed investment level.

The argument relies of course on production being deterministic and output being fully contractible. I do not claim that this is always the best approach "in reality."


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### Answer by username_3 on 2024-07-06. Score: 1

I add an answer to complement Michael Greinecker's. Unlike his answer, I consider a renegotiation design problem, whereby the two parties sign a contract prior to making investments which determines the ex-post bargaining over surplus. Hence, I endogenise the $\beta\_1$ term you describe. This sort of contract is perhaps more realistic compared to one requiring large negative penalties for potential underinvestment.

This renegotiation design problem is effectively the same setup as Agion-Dewatripont-Rey (1994 ; Econometrica). They show that although one cannot make both parties residual claimant, one can chose one party to be full residual claimant whilst simultaneously making the other $\textit{locally}$ residual claimant. These contracts are known in the literature as specific performance contracts.

A simplified version of their model is as follows. Suppose there is a buyer and a seller. The buyer and seller trade over a single indivisible good, which is valued by the buyer at $v$ and which costs the seller $c$ to provide. At the time of trading, $v$ and $c$ are observable to both parties and we allow them to bargain over the terms of trade so that trade always occurs efficiently given $v$ and $c$.

Prior to the realisation of $v$ and $c$, the buyer and seller make ex-ante investments which affect the distribution of $v$ and $c$ (and so affect the return function). Suppose there are two types, which are ordered $c\_H>v\_H>c\_L>v\_L$. Hence, the buyer and seller only trade when the realisation is $(v\_H,c\_L)$. The buyer's investment, $i$, costs $\psi(i)$, whilst the supplier's investment, $j$, costs $\phi(j)$, where $\psi,\phi$ are convex and increasing. These investments affect the distribution of types linearly: $\mathbb{P}(v=v\_H)=i$, $\mathbb{P}(c=c\_L)=j$.

The first-best level of investments maximises total welfare. $$(i^{FB},j^{FB})=\arg \max\_{i,j}\{ij(v\_H-c\_L)-\psi(i)-\phi(j)\}$$ So, the first best investment levels are implicitly defined by $$\begin{align} i^{FB}(v\_H-c\_L)&=\phi'(j^{FB}) \\
j^{FB}(v\_H-c\_L)&=\psi'(i^{FB})\end{align}$$ Absent any other contracts, there will be Williamsonian underinvestment since each party is only partially residual claimant on the ex-post surplus.

What ADR do is define a renegotiation proof contract which determines the terms of renegotiation once $v$ and $c$ are realised. This contract is signed prior to the ex-ante investment stage. The idea of the contract is to allocate all bargaining power to one party (WLOG, the buyer) whilst giving the other party the option to request a default option - specified in the contract - which that party can request regardless of whether or not it is efficient to do so. This sort of contract makes the buyer full residual claimant whilst also making the seller locally residual claimant.

Suppose the default option, as specified in the contract, is a mechanism $(\tilde{q},\tilde{p})$. After $(v,c)$ is realised, if the buyer is allocated bargaining power, it solves $$\max\_{p,q}\{qv-p\} \qquad st. \qquad p-qc\geq \tilde{p}-\tilde{q}c$$ The constraint binds at the optimum, so the problem reduces to $$\max\_{q}\{q(v-c)-\tilde{p}+\tilde{q}c\}$$ Hence, the buyer chooses $\tilde{q}=\mathbb{1}\_{\{v\geq c\}}$ so trade occurs efficiently. The buyer selects $p$ to hold the supplier down to its default option utility.

The supplier, knowing they will be held down to their default option utility, chooses their investment to solve $$\max\_j\{\tilde{p}-\tilde{q}(jc\_L+(1-j)c\_H)-\phi(j)\}$$ So, the supplier picks investment level $j^\*$ such that $$\phi'(j^\*)=\tilde{q}(c\_H-c\_L)$$

Now, $\tilde{q}$ is free for us to choose. In particular, picking $\tilde{q}$ such that $\phi'(j^{FB})=\tilde{q}(c\_H-c\_L)$ means that the supplier picks the first best level of investment! What's more is that given the supplier picks the first-best investment level, the buyer maximises the total surplus minus the supplier's surplus, solving $$\max\_{i}\{ij^{FB}(v\_H-c\_L)-\psi(i)-[\tilde{p}-\tilde{q}(j^{FB}c\_L+(1-j^{FB})c\_H)-\phi(j^{FB})]\}$$ Therefore, the buyer picks $$\psi(i^\*)=j^{FB}(v\_H-c\_L)$$ and so $i^\*=i^{FB}$. Hence, there is no problem if we allow for a sufficiently rich renegotiation design. Noldeke and Schmidt (1995; RAND) show that we can get the same result if we only consider "option contracts" which allow the supplier to choose whether or not to provide the good at a prespecified price but allows the buyer to propose new price offers. The problem with these two approaches is that they require that the supplier is able to request inefficient outcomes. Whilst this never happens in equilibrium, it is still unappealing that it is possible. ADR give some justification that their contract can be considered as a Rubinstein bargaining game where the parties can at each step request a default option, which is a more appealing implementation.

However, we really do need the renegotiation contract space to be sufficiently rich. Hart and Moore (1988 ; Econometrica) show that if specific performance contracts are not possible the only default option that a party can request is "no trade" (also known as "at-will" contracting). In this case, the first best is generally not achievable and there will be underinvestment as a result.


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