8. Incomplete Contract and Specific Investment: Hart (1995)

In this section, we consider a model in Hart (1995) with incomplete contracts. Firms have specific investment to match each other with the concern (incentive distortion) of being ditched by the other side. We will also see how property rights alleviate such distortion.

• There are two firms \(M_1\) and \(M_2\).
• Zero discount rate.
• There are two assets \(a_1\) and \(a_2\) with the owning structure
• \(M_1\) owns \(A\in\{\emptyset,\{a_1\},\{a_2\},\{a_1,a_2\}\}\)
• \(M_2\) owns \(B\in\{\{a_1,a_2\},\{a_2\},\{a_1\},\emptyset\}\)
• \(M_1\) and \(M_2\) together own both \(a_1\) and \(a_2\) without overlapping.
• \(M_1\):
• if \(M_1\) cooperates with \(M_2\), \(M_1\) buys the widget at price \(p\) from \(M_2\), and invests \(i\) at cost \(i\) to turn the widget into a surplus of \(R(i)-p\). Assume \(R(\cdot)\) is concave. So the surplus of investing \(i\) is

$$R(i)-p-i$$

• if \(M_1\) doesn't cooperate with \(M_2\), \(M_1\) buys the widget at standard price \(p_s\) from the market, and invests \(i\) to turn the widget into a surplus of \(r(i,A)-p_s\). So the surplus of investing \(i\) is

$$r(i,A)-p_s-i$$

• \(M_2\):
• if \(M_2\) cooperates with \(M_1\), \(M_2\) sells the widget at price \(p\) to \(M_1\), and puts effort \(e\) at cost \(e\) to earn a surplus of \(p-C(e)-e\), where \(C(\cdot)\) is the cost of production and \(e\) is the cost of effort. Assume \(C(\cdot)\) is convex.
• if \(M_2\) doesn't cooperate with \(M_1\), \(M_2\) sells the widget at standard price \(p_s\) to the market, and puts effort \(e\) at cost \(e\) to earn a surplus of \(p_s-c(e,B)-e\).
• Total surplus:
• When cooperation happens, the total surplus is

$$\underbrace{R(i)-p-i}_{\text{from }M_1}+\underbrace{p-C(e)-e}_{\text{from }M_2}=R(i)-C(e)-i-e$$

• When cooperation doesn't happen, the total surplus is

$$\underbrace{r(i,A)-p_s-i}_{\text{from }M_1}+\underbrace{p_s-c(e,B)-e}_{\text{from }M_2}=r(i,A)-c(e,B)-i-e$$

• Crucial assumptions:
• Cooperation is efficient:

$$R(i)-C(e)>r(i,A)-c(e,B)$$

• When cooperating, the marginal value of investment is always higher than when not cooperating; and when not cooperating, a higher asset implies a higher marginal value of investment, i.e.

$$R'(i)\ge r'(i,A)\ge r'(i,\emptyset)$$

where \(r'(i,A)\) and \(r'(i,\emptyset)\) are both w.r.t. \(i\). - Incomplete contracts: no credible agreement could be made ex-ante. - Assume equal Nash bargaining power.

Under equal Nash bargaining power (each side first keeps its outside option, then splits the extra surplus from cooperation 50-50):

• The total surplus to \(M_1\) is

$$\underbrace{r(i,A)-p_s-i}_{M_1\text{ outside option}}+\frac12\underbrace{\big[(R(i)-C(e))-(r(i,A)-c(e,B))\big]}_{\text{total surplus}}$$

$$\Leftrightarrow r(i,A)-p_s+\tfrac12 R(i)-\tfrac12 C(e)-\tfrac12 r(i,A)+\tfrac12 c(e,B)-i-e$$

$$\Leftrightarrow \tfrac12 r(i,A)-p_s+\tfrac12 R(i)-\tfrac12 C(e)+\tfrac12 c(e,B)-i-e$$

• The total surplus to \(M_2\) is

$$\underbrace{p_s-c(e,B)-e}_{M_2\text{ outside option}}+\frac12\underbrace{\big[(R(i)-C(e))-(r(i,A)-c(e,B))\big]}_{\text{total surplus}}$$

$$\Leftrightarrow p_s-c(e,B)+\tfrac12 R(i)-\tfrac12 C(e)-\tfrac12 r(i,A)+\tfrac12 c(e,B)-i-e$$

$$\Leftrightarrow p_s+\tfrac12 R(i)-\tfrac12 C(e)-\tfrac12 r(i,A)-\tfrac12 c(e,B)-i-e$$

Given cooperation, the first-best level of investment maximizes social surplus:

$$\max_i\ \underbrace{R(i)-C(e)-i-e}_{\text{total surplus when cooperate}}$$

• The f.o.c. w.r.t. \(i\):

$$R'(i^\star)=1$$

• The f.o.c. w.r.t. \(e\):

$$C'(e^\star)=-1$$

• \(i^\star\) and \(e^\star\) denote the first-best levels.

We say that there is a hold-up problem when one party won't input its best effort in fear that the other party will take advantage of that effort.

• Consider the maximization problem of \(M_1\) (optimizing \(M_1\)'s payoff from §8.2 over \(i\)):

$$\max_i\ \tfrac12 r(i,A)-p_s+\tfrac12 R(i)-\tfrac12 C(e)+\tfrac12 c(e,B)-i-e$$

• The f.o.c. w.r.t. \(i\) is

$$\frac12 r'(\hat i,A)+\frac12 R'(\hat i)=1 \tag{8.1}$$

• Since \(R(\cdot)\) is concave and \(R'(\hat i)\ge r'(\hat i,A)\), plugging into (8.1) gives

$$R'(\hat i)\ge 1=R'(i^\star)\ \Rightarrow\ \hat i\le i^\star$$

(because \(1=\tfrac12 r'(\hat i,A)+\tfrac12 R'(\hat i)\le R'(\hat i)\); then by the concavity of \(R\), i.e. \(R'\) decreasing, \(\hat i\le i^\star\).)

• Consider the maximization problem of \(M_2\):

$$\max_e\ p_s+\tfrac12 R(i)-\tfrac12 C(e)-\tfrac12 r(i,A)-\tfrac12 c(e,B)-i-e$$

• The f.o.c. w.r.t. \(e\) is

$$\frac12 c'(\hat e,B)+\frac12 C'(\hat e)=-1 \tag{8.2}$$

• Since \(C(\cdot)\) is convex and \(c'(\hat e,B)\ge C'(\hat e)\), plugging into (8.2) gives

$$C'(\hat e)\le -1=C'(e^\star)\ \Rightarrow\ \hat e\le e^\star$$

(because \(-1=\tfrac12 c'(\hat e,B)+\tfrac12 C'(\hat e)\ge C'(\hat e)\); then by the convexity of \(C\), i.e. \(C'\) increasing, \(\hat e\le e^\star\).)

• So, in the hold-up problem, \(M_1\) invests less than (under-investment) the first-best level, and \(M_2\) also exerts less effort than the first-best level.

Below are some cases where the optimal owning structure might differ.

• When assets \(a_1\) and \(a_2\) are independent, i.e.

$$r'(i,\{a_1,a_2\})=r'(i,\{a_1\})$$

• \(a_2\) is not helping \(M_1\) at all;
• but \(a_2\) could help \(M_2\) raise its effort value \(\hat e\) to be closer to the first best \(e^\star\);

• so, optimally, \(A=\{a_1\}\) and \(B=\{a_2\}\) (split ownership).

• When assets \(a_1\) and \(a_2\) are strictly complementary, i.e.

$$r'(i,\{a_1\})=r'(i,\emptyset)$$

• without \(a_2\), we see that \(a_1\) is not helping \(M_1\) at all;
• so, optimally, we should put \(a_1\) and \(a_2\) together in one firm to approach the first-best outcome;
• whether it's firm \(M_1\) or firm \(M_2\) depends on which firm is more crucially affected by asset ownership.