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A.1

Proof of proposition 1.

The optimal rating qualityq ₂[0;1]is derived by the …rst order condition:

(1 p)m=c0(q ) (A1.)

E¤ect of p: As p increases, the left hand side of A1. becomes smaller. To keep equality, the right hand side has to become smaller, too. Due to the characteristics of the cost functionc(:), the optimal rating quality q has to decrease.

E¤ect of m: m increases, if d increases and/or b decreases. The left hand side

increases, therefore the optimal rating quality q has to increase, too.

E¤ect ofc0(q ): A more e¢ cient rating technology is expressed with lower costs for a given rating standard. If rating quality increases, the rise of costs is lower for a more e¢ cient technology: c0_high(q)< c0_low(q). The marginal rating costs for a given rating quality is lower, if cost e¢ ciency becomes higher. The value of left hand side is exogenous and constant. If cost e¢ ciency increasesq , the rating standard has to rise in order to keep the right hand side constant.

A.2

Proof of proposition 2.

The optimal rating standard if rating quality q^ is unobservable and with costly regulation is derived by the …rst order condition:

As the left hand side of (A2.) is the same as the left hand side of (A1.), we have:

c0(q ) =c0(^q) +e0(^q) (A.2’)

Sincec(:) ande(:) are increasing and convex inq, thereforeq has to be larger than

^

q: q >q.^

For the proof of parts (i) to (iii) of proposition 2 see the proof of proposition 1. Part (iv) claims that the optimal rating standard is higher, if optimal penalty^

increases. An example for a rise in^ is an increase in the pro…ts per rating due to market power in the CRA sector. The necessary e¤ort that has to be employed in order to enforce a given rating standard, is given by:

e(^q) =h 1 c(^q)

^ (A3.)

If^increases, the optimal e¤ort corresponding to a given rating standard decreases. From this follows that the marginal e¤orte0_{(:)decreases either. The right hand side}

therefore becomes smaller for a given rating standard. Due to the characteristics of the functions c(:) and e(:), the optimal rating standard q^must increase to ensure equality of equation (A.3).

A.3

Proof of proposition 3.

Why the optimal rating standard in the presence of joint provision of services (q) and~

analogously in the case of a direct bribe, is lower than the optimal rating standard in 3.3 (q), can be shown as follows: The …rst-order condition of the regulator’s decision^

problem is given by

(1 p)m=c0(~q) +e0(~q) (A.4) The left hand side of A4. the same as in equation A.2, therefore it must be true that:

Since the e¤ects are analog for the direct ( ) and the indirect bribe ( ), the proof is presented for > 0 only. Taking into account the potential bribe , the necessary e¤ort to enforce a given rating standard q is larger in the case of joint provision of services:

e(q) =h 1 c(q)

^ < e(q) =h

1 c(q) +

^

The marginal e¤ort with bribe is therefore larger than the marginal e¤ort without bribe for a given rating quality. Therefore the optimal rating standard has to be lower in the case of joint provision of services, to ensure equality of equation (A.4’), given the characteristics ofc(:)and e(:): q <~ q.^

That the optimal rating standard is decreasing as the rent of the joint provision of services increases can be shown by total di¤erentiating equation (A.4) and using e0_{(~q) = h}0 1 c(~q)+ ^ c0_(~q) ^ : d(1 p)m | {z } =0 =c00(~q)dq~+e00(~q)dq~+h00 1 c(~q) + ^ c0_(~q) ^2 d ! _ddq~ = h00 1 c(~q)+ ^ c0_(~q) ^2 c00_{(~q) +e}00_(~q) <0 (A.5)

Equation is smaller than zero, because of the convexity of e(:), c(:), and h 1_().

A.4

Proof of proposition 4.

The social value with joint provision of services is given by V~, whereas the social value with rating only is given by V^. If = 0, then V~ = ^V. Equation (A.5) showed that _ddq~ <0. Therefore, given the characteristics ofc(:), it must be true that

@c0_(~q)

@ < 0. Above, we have shown that e0(~q) = h0

1 c(~q)+ ^

c0_(~q)

^ . Di¤erentiating with respect to yields @e_@0(~q) =h00 1 c(~q)+

^

c0_(~q)

^2 >0. The partial derivative of

the social value V~ and V^ with respect to is given by: _@@V~ = _c01_(~q)(c0(~q) e0(~q))

and _@@V^ = 0. As increases, the di¤erence (c0_{(~q) e}0_{(~q)) decreases, since c}0

the social value V~ decreases further as increases. Therefore, it is optimal to prevent the joint provision of services for any >0, since V <~ V^.

A.5

Proof of proposition 5.

If _@@V~ > 0 at = 0, then we must have c0(~q) > e0(~q). From this follows that

~

V increases, if becomes larger, as long as is relatively small. Given the characteristics of @c_@0(~q) and @e_@0(~q), V~ decreases, for relatively large values of . Therefore, at some critical level = , we have V~ = ^V. For values < , we have V >~ V^, and therefore allowing the joint provision of services is optimal. For values > , we haveV <~ V^, and therefore forbidding the joint provision of services is optimal.