Caracterización del Modelo de Gestión Humana de la IES

Absent any information asymmetry, all issuers are able to sell their issues in the market for their true values. Since we assumed firms which invest in the projects issue debts of unit face value, the capital raised by one such firm is equal to q. Then the expected profit of firm x, provided that all firmsy < x have invested, is given by

qqR(qx)−qc−q=0

The left hand side of equation (2.1) is decreasing inxand is negative for large enoughx by assumption 2.1. Further, assumption 2.1 ensures that the left hand side of equation (2.1) is strictly positive forx= 0. Therefore, there exists a unique xFB >0 that solves equation (2.1). Firm xFB > 0 is the firm which is indifferent between investing and not, provided that all firmsy < xFB invest. Another way to consider the first best solution is to look for

the aggregate investment in the projects IFB for which an extra unit of investment yields

an expected return of 0. Clearly the first best investment level, IFB_{, is equal toqx}FB_{, and}

therefore we can rewrite equation 2.1 as

qR(IFB) =c+ 1 (2.2)

Condition (2.2) equates the expected return of project IFB (the left hand side), to a firm’s cost c of investing in a project plus the cost of capital, which is equal to 1 since investors are competitive by assumption. Note that the first best solution is purely determined by the projects expected returnqR(.) and does not react to changes inq and R(.) as long as the expected return remains constant. This is not the case for the equilibrium solution, however, as it will be demonstrated in the next section. We will utilize this property of the first best solution in our comparative statics analysis in later sections.

2.5. Equilibrium

We start by giving the formal definition of an equilibrium of the investment game.

Definition 2.1. An equilibrium of the investment game is a set of firms’ investment deci- sions, the CRA’s and issuers’ strategies, and issue prices such that

• Given the strategy of the CRA and issuers, and prices, any profitable investment in the projects is undertaken.

• Given firms’ investment decisions, issuers’ strategies, and prices, the CRA’s profit is maximized.

• Provided the rating technology and fee, issuers maximize the capital raised.

• Given the investment decisions and the CRA and issuers’ strategies, uninformed in- vestors’ prices for issues offered in the market is set to their corresponding expected values. The expectations are calculated over uninformed investors’ beliefs, derived by Bayes rule whenever possible.

We now proceed to derive an equilibrium of the investment game. We start by taking firms’ investment decisions as given and solve the equilibrium of the rating sub-game. Note that since the face value of debts issued by all investing firms is equal to 1, then the total face value of debts issued by investing firms, m2, is equal to the measure of firms which

invest in the projects. Then in the rating sub-game, there are 3 issuer types v1, v2, v3 and

a total of m1, m2, m3 face value of each type of issue, respectively. This rating sub-game2

has a unique equilibrium in which the rating system belongs to table 2.1, described by the following lemma.

Lemma 2.1. There exists an equilibrium of the rating sub-game in which the CRA setsp=

δ(m3+m2)

m3+δm2 andφ=µ(1−δ)

m3v3+m2v2

m3+m2 . Typev1 issuers do not solicit ratings. Market prices of unrated issues, as well ass1rated ones, are set to 0. Typev2 andv3issuers do solicit ratings.

Conditional on rating s3, price of types v2 and v3 is set to U3 = (1−δ)m3_mv3₃+₊m_m2₂v2 +δv3.

Conditional on rating s2, price of type v2 is set to v2. All issues are traded at set market

prices.

Ratings enable issuers to profit by selling their assets to uninformed investors who demand lower rates (0 in our model) compared to informed investors. The CRA benefits by extract- ing a fraction of these profits. Lemma 2.1 states that the optimal strategy of the CRA in the rating sub-game is to choose a noisy rating system. The intuition behind this result is that in order for the CRA to maximize profit, that is, to charge the highest possible φ, it needs to choose rating precisionp such that the willingness to pay for ratings of all the targeted

types are equal. In the equilibrium of Lemma 2.1, only types v2 and v3 are targeted. By

2

assigning ratings3 to type v2 issues with some non zero probability, the willingness to pay

of these two types are made equal.

Since we made the assumption that firms have no capital of their own, rating fees must naturally be paid by the proceedings of the issues. Consequently, the funds raised by all issuers who solicit a rating must be at least as large as the rating fee, regardless of the ratings issued. Among all the rated issue types, the smallest sale proceedings is equal to

v2, which is the outcome when a type v2 is rated s2. Therefore, imposing the constraint

v2 > φ ensures that all rated types are able to pay the rating fee. A sufficient condition

that ensures v2 > φis

Lemma 2.2. µ(1−δ)< _vq

3 ⇒v2 > φ

In making their investment decisions, firms take into account the impact of total investment on the CRA’s choice of ratings. Suppose at first that no firm is investing in the projects, that ism2= 0, and consider the incentives of firm 0. Assumption 2.1 ensures that investing

in project 0 is profitable. To see this notice that if firm 0 decides to raise funds by selling bonds to uninformed investors, the amount raised net of the rating fee is at least as large

as the amount offered by informed investors which is equal to δq. We can then conclude

that the measure of firms investing in the projects in equilibrium must be positive. Now suppose all firms x < m2 have decided to invest while firms x > m2 have decided not to

invest and consider the incentives of firm m2. If firmm2 invests, the expected return per

unit of capital invested is equal toqR(I), where the aggregate investment of all firms in the projects,I, is equal to I =m2[pv2+ (1−p)U3−φ] =m2 (1−µ)(1−δ)m3v3+m2q m3+m2 +δq , (2.3)

where the last equality follows directly from Lemma 2.1. The cost of investing a unit by firmm2, on the other hand, is the sum ofcand the expected cost of capital, which is equal

to c+ v2 pv2+ (1−p)U3−φ =c+m2q I =c+ q (1−µ)(1−δ)m3v3+m2q m3+m2 +δq

For the measure of firms investing in the equilibrium outcome to be equal to m2, firm m2

must be indifferent between investing and not investing, that is

qR(I) =c+ q

(1−µ)(1−δ)m3v3+m2q

m3+m2 +δq

. (2.4)

Asm2increases, the total amount invested in the projects,I, also increases which results in

a decrease in the expected return of firmm2’s investment. At the same time, the right hand

side of the indifference condition 2.4 is increasing in m2 since q =v2 < v3 by assumption.

As a result, there exist a uniquem2 for which the indifference condition (2.4) is satisfied.

Proposition 2.1. The investment game has a unique equilibrium in which the CRA’s rating technology belongs to table 2.1. The equilibrium measure of firms investing in the projects

mEq₂ and the aggregate investment IEq are the unique solutions to the following system of equations.      IEq = mEq₂ h(1−µ)(1−δ)m3v3+m2q m3+m2 +δq i qR(IEq) = c+m Eq 2 q IEq

A comparison of the first best and equilibrium indifference conditions (2.2) and (2.4) reveals how ratings can influence firms investment decisions through their impact on firms’ effective cost of capital. In general, the cost of capital in equilibrium, m

Eq 2 q

IEq will not be equal to 1. Consequently, aggregate investment in equilibrium diverges from first best solution. For

a given project expected returns, however, we can find a q for which the two solutions

Proposition 2.2. Keep project expected returns, qR(.) constant. Then for µ < 1

1+_mIFB

3v3 , there exists a uniqueq∗ for which the equilibrium outcome and the first best coincide.

q∗=(1−µ)m3v3−µI

FB m3

.