Appendix B (Sufficient conditions for (

1

Learning from Failures: Optimal Contracts for Experimentation and Production Khalil, Lawarree, and Rodivilov

Supplementary Appendix

This document contains Appendices B, C, D, E, F, G

Appendix B (Sufficient conditions for (𝐼𝐶

) to be binding) Claim B: There exist 𝜷

< 𝟏 and 𝝀

𝑯

(𝝀

) > 𝝀

for any 𝝀

> 𝝀 such that (𝑰𝑪

) is binding if 𝜷

> 𝜷

and 𝝀 < 𝝀

< 𝝀

< 𝝀

𝑯

(𝝀

).

Proof: We prove by contradiction that if 𝛽 is high enough, and 𝜆 and 𝜆 are high enough and not too far apart from each other, both (𝐼𝐶) are binding. In Step 1, we consider values of 𝜆 and 𝜆 such that the first-best stopping times are 𝑇 > 𝑇 . In Step 2, we assume that only (𝐼𝐶

) is binding, and, as a result, only the stopping time for the low type is distorted (𝑇 = 𝑇 and 𝑇 = 𝑇 ). We show that if (𝐼𝐶

) is ignored, then the equilibrium value of 𝑇 is strictly larger than 𝑇 if 𝛽 is large enough. Finally, in Step 3, we show when 𝜆 and 𝜆 are close enough to each other and 𝛽 is large enough, the two (𝐼𝐶) constraints require 𝑇 ≤ 𝑇 . This implies a contradiction, and that both (𝐼𝐶) constraints must be binding.

Step 1: The first-best stopping times (𝑻

> 𝑻

). Consider values of 𝜆 and 𝜆 , where 𝜆 <

𝜆 < 𝜆 , such that 𝑇 > 𝑇 . Recall that the first-best stopping time is non-monotonic in 𝜆 : there exists 𝜆 ∈ (0,1), such that > 0 for 𝜆 < 𝜆 and ≤ 0 for 𝜆 ≥ 𝜆 (see Claim 1 in Supplementary Appendix F).

Step 2: Stopping times if the (𝑰𝑪

) is ignored (𝑻

> 𝑻

). We will show that if 𝜆 and 𝜆 are

close enough, and 𝛽 is high enough, then the first best order of termination dates is preserved in

the second best when (𝐼𝐶

) is ignored. Recall that if only the (𝐼𝐶

) is binding, then the

stopping time for the high type is not distorted (𝑇 = 𝑇 ), and the distortions in 𝑇 are

determined by the following F.O.C. (see part II of Appendix A):

2

= 0,

where 𝑈 (𝑇 ) = 𝛿 𝑃 ∆𝑐 𝑞 𝑐 is the rent of the high type.

We next prove that, for any 𝑇 , the high-type’s rent 𝑈 (𝑇 ) decreases with 𝛽 and goes to zero as 𝛽 converges to 1. Therefore, for large values of 𝛽 , the equilibrium value of 𝑇 will remain close to its first-best value 𝑇 .

First, we prove that ∆𝑐 is decreasing in 𝛽 as it plays a critical role in the rent

expression. We rewrite ∆𝑐 =

(1 − 𝛽 )𝛽 𝑐 − 𝑐 =

( )

𝑐 − 𝑐 (1 − 𝜆 ) − (1 − 𝜆 ) . Note that this expression above goes to zero as 𝛽 converges to 1.

Consider the derivative of

( )

with respect to 𝛽 :

⎣

⎢⎢

⎢

⎡

( )

⎦

⎥⎥

⎥

⎤

=

( ) ( )

( )

=

( ) ( )

( )

=

( ( ) ( ) ) ( )

( )

=

( ( ) ( ) ) ( )

( )

=

1

We could have also made our argument for Step 2 relying on small values of 𝜈.

3

( )

( )

< 0 if 𝛽 >

,

for 𝑡 > 𝑇 . In Step 3, we will consider values for 𝜆 and 𝜆 such that both 𝑇 and 𝑇 are smaller than 𝑡 , where 𝑡 is a unique time period 𝑡

such that ∆𝑐 achieves the highest value at this time period, which is formally defined as

𝑡

= 𝑎𝑟𝑔 max (1 − 𝜆 ) − (1 − 𝜆 )

(1 − 𝛽 + 𝛽 (1 − 𝜆 ) )(1 − 𝛽 + 𝛽 (1 − 𝜆 ) ) .

Second, when 𝛽 >

, note that 𝑃 ∆𝑐 also decreases with 𝛽 since 𝑃 and

∆𝑐 are decreasing in 𝛽 .

Third, consider the remaining term 𝑞 𝑐 that depends on 𝛽 . Recall, when only (𝐼𝐶

) is binding, the distortion in 𝑞 𝑐 is determined by (see part III of Appendix A)

𝑉 𝑞 𝑐 = 𝑐 +

∆𝑐 .

Therefore, the distortion in 𝑞 𝑐 from 𝑞 𝑐 becomes smaller for any 𝑇 as 𝛽 increases since ∆𝑐 is decreasing in 𝛽 when 𝛽 >

( )^𝑡Δ

.

Thus in step 2, we proved that, for any 𝑇 , the high-type’s rent 𝑈 (𝑇 ) decreases with 𝛽 and goes to zero as 𝛽 converges to 1. Define 1 > 𝛽 >

such that 𝑇 > 𝑇 for 𝛽 >

𝛽 . Intuitively, for large values of 𝛽 , the equilibrium value of 𝑇 will remain close to its first- best value 𝑇 . Given that 𝑇 > 𝑇 , we will have 𝑇 > 𝑇 for high values of 𝛽 .

2

Note that we ignored the term 𝛿 in the rent expression as the incentive is to increase 𝑇 , which will only help

preserve the first-best order, 𝑇 > 𝑇 .

4

Step 3: The (𝑰𝑪

) is violated if 𝑻

> 𝑻

. We will show that if 𝜆 and 𝜆 are close enough, and 𝛽 is high enough, assuming that (𝐼𝐶

) is slack when deriving the second best contract leads to a contradiction.

We start by identifying the restriction on 𝑇 and 𝑇 imposed by the (𝐼𝐶) constraints below:

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥

≥ 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 − ∆𝑐 𝑞 𝑐 ,

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥

≥ 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 + ∆𝑐 𝑞 𝑐 .

Since we proved that in the optimal contract (i) there is no restriction on when to reward the high type when only the (𝐼𝐶

) is binding, and (ii) the high type agent is rewarded in the very first period when both types’ (𝐼𝐶) are binding, it is without loss of generality to consider

𝑦 > 0 = 𝑥 = 𝑦 for all 𝑡 > 1.

Then, from the binding (𝐼𝐶

) evaluated at 𝑦 = 𝑥 = 0 for all 𝑡 ≥ 1, we obtain

𝑦 =

.

Replacing 𝑦 with the expression above in (𝐼𝐶

) and plugging 𝑦 = 𝑥 = 0 for all 𝑡 ≥ 1, we have

(𝐼𝐶

) 0 ≥ 𝛽 𝛿𝜆

+ 𝛿 𝑃 0 − ∆𝑐 𝑞 𝑐 ,

𝛿 𝜆 𝑃 ∆𝑐 𝑞 𝑐 ≥ 𝛿 𝑃 𝜆 ∆𝑐 𝑞 𝑐 ,

which can be rewritten as

∆𝑐 ≥ ∆𝑐 .

To focus on the screening role of expected cost, output, and probability of failure, we will ignore the ratio by assuming that 𝛿 is large enough.

Next, we will show that when 𝜆s are close to each other, the Δ𝑐 is increasing in 𝑡. Then,

at the second best stopping times that were derived ignoring (𝐼𝐶

), we have ∆𝑐 < ∆𝑐

5

since 𝑇 > 𝑇 = 𝑇 (see Step 2). Thus, if the ratio ≤ 1, which we will show in the later part of the proof, we would have a contradiction as it would violate (𝐼𝐶

).

Recall that the difference in the expected cost, ∆𝑐 , is a non-monotonic function of time:

initially increasing and then decreasing, reaching a maximum at time period 𝑡 . Formally,

∆𝑐 = 𝑐 − 𝑐 = 𝛽 𝑐 + (1 − 𝛽 ) 𝑐 − 𝛽 𝑐 + (1 − 𝛽 )𝑐

= 𝑐(𝛽 − 𝛽 ) − 𝑐(−1 + 𝛽 + 1 − 𝛽 ) = (𝛽 − 𝛽 ) 𝑐 − 𝑐

= 𝛽 (1 − 𝜆 )

𝛽 (1 − 𝜆 ) + (1 − 𝛽 ) − 𝛽 (1 − 𝜆 )

𝛽 (1 − 𝜆 ) + (1 − 𝛽 ) 𝑐 − 𝑐

=

( ) ( ( ))

( ( ) ( )) ( ) ( )

𝑐 − 𝑐

= 𝛽 (1 − 𝛽 ) 𝑐 − 𝑐

.

We show next that for any 𝜆 there exists a value of 𝜆 , called 𝜆 (𝜆 ) such that if 𝜆 <

𝜆 < 𝜆 (𝜆 ) both 𝑇 and 𝑇 are on the increasing part of ∆𝑐 : 𝑇 , 𝑇 < 𝑡

. Intuitively, when 𝜆s are close to each other, the 𝛥𝑐 function is relatively flat and both 𝑇 and 𝑇 are on the

increasing part of ∆𝑐 : 𝑇 , 𝑇 < 𝑡

. First, note that

> 0, which means 𝛥𝑐 increases with 𝜆 .

Second, we prove that

< 0.

The F.O.C. determining 𝑡

is

( ( ) ) ( ( ) )

−

( ( ) ) ( ( ) )

= −

ln(1 − 𝜆 ) +

6

(1 − 𝛽 + 𝛽 (1 − 𝜆 ) )(1 − 𝛽 + 𝛽 (1 − 𝜆 ) )(1 − 𝜆 )

(1 − 𝛽 + 𝛽 (1 − 𝜆 ) ) (1 − 𝛽 + 𝛽 (1 − 𝜆 ) ) ln(1 − 𝜆 )

= −

( ( ) ) ( ( ) )

ln(1 − 𝜆 ) +

( ( ) ) ( ( ) )

ln(1 − 𝜆 ) = 0.

Thus, 𝑡

is the value of 𝑡 such that

𝑡

:

ln(1 − 𝜆 ) =

ln(1 − 𝜆 ).

Therefore, from the Implicit Function Theorem, we have

∆

= − ,

where

Φ =

ln(1 − 𝜆 ) −

( ( ) )

ln(1 − 𝜆 ).

We next prove that < 0 and < 0, in turn.

First, =

=

( )

( )

( ( ) )

+

( )

( ( ) )

=

( ( ) )

+

( )

( ( ) )

=

+

7

=

1−𝜆^{𝐻 𝑡−2} 𝛽₀(𝑡−1) 1−𝛽₀+𝛽₀ 1−𝜆^{𝐻 𝑡−1} +2𝛽₀(𝑡−1) 1−𝜆^{𝐻 𝑡−1} ln 1−𝜆^𝐻 − 1−𝛽₀+𝛽₀ 1−𝜆^{𝐻 𝑡−1}

1−𝛽₀+𝛽₀ 1−𝜆^{𝐻 𝑡−1}

3

< 0.

Second, since

⎣⎢

⎢⎢

⎡

⎦⎥

⎥⎥

⎤

=

( ( ) )

=

= ,

we have

=

[ln(1 − 𝜆 )] −

( ( ) )

[ln(1 − 𝜆 )] ,

which given the F.O.C. for 𝑡

:

( ( ) )

ln(1 − 𝜆 ) =

ln(1 − 𝜆 ), can be rewritten as

=

−

ln(1 − 𝜆 ).

Next,

( ( ) )

−

=

=

−

8

( ) ( ) ( )

( ( ) )( ( ) )

=

> 0.

Thus,

=

( ( ) )( ( ) )

ln(1 − 𝜆 ) < 0.

Therefore,

∆

= − < 0.

Thus, when 𝜆s are close to each other both 𝑇 and 𝑇 are on the increasing part of ∆𝑐 : 𝑇 , 𝑇 < 𝑡

.

We next prove that ≤ 1 if 𝜆s are close to each other and 𝛽 is high enough.

First, consider the ratio of outputs . Recall, when only (𝐼𝐶

) is binding, the distortions in 𝑞 𝑐 and 𝑞 𝑐 are determined by

𝑉 𝑞 𝑐 − 𝑐 =

∆𝑐 > 0 and

𝑉 𝑞 𝑐 − 𝑐 = 0, respectively,

In Step 2 above, we proved that the distortion in 𝑞 𝑐 from 𝑞 𝑐 becomes smaller for any 𝑇 as 𝛽 increases because ∆𝑐 is decreasing in 𝛽 . Thus, as 𝛽 increases, the ratio

converges to , where

𝑉 𝑞 𝑐 = 𝑐 and 𝑉 𝑞 𝑐 = 𝑐 .

9

Since 𝑇 > 𝑇 in the second best derived ignoring (𝐼𝐶

) (see Step 2), the ratio

≤ 1 because 𝑐 < 𝑐 when 𝜆s are close to each other. Define 𝜆 (𝜆 ) < 1 such that 𝑐 < 𝑐 for 𝑇 > 𝑇 , when 𝜆 < 𝜆 (𝜆 ), where 𝑇 is derived ignoring (𝐼𝐶

) and 𝑇 = 𝑇 (see Step 2).

Intuitively, when 𝜆s are close to each other both types have similar expected cost after the same amount of failures.

Second, we prove that < 1 if 𝛽 is high enough:

( )

< 1,

(1 − 𝛽 )𝜆 +𝜆 𝛽 (1 − 𝜆 ) < (1 − 𝛽 )𝜆 +𝜆 𝛽 (1 − 𝜆 ) , (1 − 𝛽 )(𝜆 − 𝜆 ) < 𝛽 𝜆 (1 − 𝜆 ) − 𝜆 (1 − 𝜆 ) , (𝜆 − 𝜆 ) < 𝛽 𝜆 (1 − 𝜆 ) − 𝜆 (1 − 𝜆 ) + (𝜆 − 𝜆 ) ,

𝛽 >

( ) ( ) ( )

.

Replacing 𝑇 by 𝑡

and 𝑇 by 1, the condition becomes

𝛽 > 𝛽 ≡

.

Therefore, if (1) 𝛽 is high enough, such that 𝛽 > 𝛽 ≡ max 𝛽 , 𝛽 , and (2) 𝜆s are close to each other, which is formally described as, for any 𝜆 there exists a value of 𝜆 , called 𝜆 (𝜆 ) ≡ min 𝜆 , 𝜆 such that 𝜆 < 𝜆 < 𝜆 (𝜆 ), then the (𝐼𝐶

) is violated. Thus, assuming that only (𝐼𝐶

) is binding leads to a contradiction.

This concludes the proof of Claim B.

Q.E.D.

10

Appendix C (Ex post moral hazard)

As before, the (𝐼𝐶

) will be binding. The (𝐼𝐶

) may again be binding (Case 1) or slack (Case 2). Since the two (𝐸𝑀) constraints for the low type imply that the low type will receive rent due to ex post moral hazard, this rent will make it less attractive for the low type to lie. We prove that it might still be optimal to reward the low type for failure and late success (see Case 1 below). In particular, in Lemma B.1 we prove that if a limited liability constraint is binding when success is public then the corresponding ex post moral hazard constraint must be binding as well. This should appear intuitive: in the optimal contract, the agent is paid the additional ex post moral hazard rent at the same periods as in Case B of Appendix A.

We assume that success is privately observed by the agent, and that an agent who finds success in some period 𝑗 can choose to announce or reveal it at any period 𝑡 ≥ 𝑗. Thus, we assume that success generates hard information that can be presented to the principal when desired, but it cannot be fabricated.

The 𝐸𝑀𝐻 constraint makes it unprofitable for the agent to hide success in the last period. Note in 𝐸𝑀𝐻 that the incentive to hide is due to 𝑥 only. The agent will be

undercompensated for suppressing success as 𝑐 − 𝑐 < 0. The 𝐸𝑀𝑃 constraint makes it unprofitable to postpone revealing success. The two together imply that the agent cannot gain by postponing or hiding success.

𝐸𝑀𝐻 𝑦 ≥ 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 for 𝜃 = 𝐻, 𝐿, and 𝐸𝑀𝑃 𝑦 ≥ 𝛿𝑦 for 𝑡 ≤ 𝑇 − 1.

We start by proving that (𝐼𝐶

) is binding and that the (𝐸𝑀𝐻 ), (𝐸𝑀𝑃 ), and (𝐿𝐿𝑆 ) are slack. In the main model with public success, the high type was either rewarded for the very first success (Case B), or there were no restrictions on when to pay the high type (Case A).

Therefore, it is without loss of generality to consider a contract for the high type where 𝑦 > 0 = 𝑥 = 𝑦 for 𝑡 > 1.

Note that with 𝑦 > 0 = 𝑥 = 𝑦 for 𝑡 > 1, the ex post moral hazard constraints for the high type are automatically satisfied:

(𝐸𝑀𝐻 ) 𝑦 = 0 > 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 = 𝑐 − 𝑐 𝑞 𝑐 , and

(𝐸𝑀𝑃 ) 𝑦 = 0 = 𝛿𝑦 = 0 for 𝑡 ≤ 𝑇 − 1.

With the described payment scheme for the high type it must be that (𝐼𝐶

) constraint is binding because otherwise 𝑦 could be lowered without violating any other constraint. Finally, (𝐿𝐿𝑆 ) are automatically satisfied for 𝑡 ≤ 𝑇 − 1 due to (𝐸𝑀𝑃 ).

We first characterize the optimal payment structure, 𝑥 , {𝑦 } , then the optimal length of experimentation, 𝑇 and 𝑇 , and finally the optimal outputs {𝑞 (𝑐)} , 𝑞 𝑐 ,

{𝑞 (𝑐)} and 𝑞 𝑐 .

11

I. Optimal payment structure, 𝒙

, {𝒚

}

, and 𝒚

We will show that the solution to the principal’s optimization problem depends on which combination of constraints is binding. We explore each case separately in what follows. While we know 𝑥 = 0 in the optimal contract, it is convenient to present the principal’s problem retaining 𝑥 in order to facilitate the comparison with the public success case.

Recall that the expected surplus net of costs for 𝜃 = 𝐻, 𝐿 by Ω 𝜛 =

𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) − 𝛤 + 𝛿 𝑃 𝑉 𝑞 𝑐 −

𝑐 𝑞 𝑐 − 𝛤 . The principal’s optimization problem then is to choose contracts 𝜛 and 𝜛 to maximize the expected net surplus minus rent of the agent, subject to the constraints given below:

𝑀𝑎𝑥 𝐸 Ω 𝜛 − 𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑦 − 𝛿 𝑃 𝑥 subject to:

(𝐼𝐶

) 𝛽 𝛿𝜆 𝑦 + 𝛿 𝑃 𝑥

= 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 + ∆𝑐 𝑞 𝑐 ,

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥

≥ 𝛽 𝛿𝜆 𝑦 + 𝛿 𝑃 𝑥 − ∆𝑐 𝑞 𝑐 ,

𝐿𝐿𝐹 𝑥 = 0, 𝐿𝐿𝐹 𝑥 ≥ 0, 𝐿𝐿𝑆 𝑦 ≥ 0,

(𝐸𝑀𝑃 ) 𝑦 ≥ 𝛿𝑦 for 𝑡 ≤ 𝑇 − 1, and

(𝐸𝑀𝐻 ) 𝑦 ≥ 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 for 𝜃 = 𝐻, 𝐿.

Case 1: Both the (𝑰𝑪

) and (𝑰𝑪

) constraints bind.

Using the binding (𝐼𝐶

) and (𝐼𝐶

) constraints we express x and 𝑥 as follows:

0 = 𝑥 𝑦 , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐

= +

+

∆ ∆

;

12

𝑥 𝑦 , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐

= +

+

∆ ∆

.

Labeling 𝜉 , 𝜉 , 𝜔 , {𝛼 } , {𝛼 } as the Lagrange multipliers of the corresponding constraints, the optimal contract has the following Lagrangian:

ℒ = 𝐸

Ω

( 𝜛

) − 𝛿

𝑃

𝑥

𝑦

, 𝑦

𝑡=1 𝑇^𝐿

, 𝑇

, 𝑇

, 𝑞

𝑐

𝐻

, 𝑞

𝑐

−𝛽

∑

𝛿

1 − 𝜆

𝜆

𝑦

+𝜉 𝑥 𝑦 , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐

+𝜉 𝑥 𝑦 , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 + ∑ 𝛼 (𝑦 − 𝛿𝑦 ) + 𝜔 𝑦

+𝛼 𝑦 − 𝑥 𝑦 , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 − 𝑐 − 𝑐 𝑞 𝑐 .

The relevant Kuhn-Tucker conditions with respect to 𝑦 are:

𝜕ℒ

𝜕𝑦 = −(1 − 𝜐) 𝛽 𝛿 (1 − 𝜆 ) 𝜆 + 𝛿 𝑃 𝛽 𝛿 𝑃 (1 − 𝜆 ) 𝜆 − 𝑃 (1 − 𝜆 ) 𝜆

𝛿 𝑃 𝑃 − 𝑃 𝑃

−𝜐𝛿 𝑃 + 𝛼 1{𝑡 < 𝑇 } − 𝛿𝛼 1{𝑡 > 1} +

𝛼 1{𝑡 = 𝑇 } 1 − + 𝜔 1{𝑡 = 𝑇 }

+𝜉 + 𝜉 .

We can rewrite the Kuhn-Tucker conditions above as follows:

ℒ

= 𝑃 𝑓 (𝑡) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (𝑡) + 𝛼 1{𝑡 < 𝑇 }

−𝛿𝛼 1{𝑡 > 1}

+𝛼 1{𝑡 = 𝑇 } 1 − + 𝜔 1{𝑡 = 𝑇 } = 0.

13

Lemma B.1. Lagrange multipliers corresponding to the (𝐸𝑀𝑃 ) for 𝑡 ≤ 𝑇 − 1 and

(𝐸𝑀𝐻 ) constraints {𝛼 } , for a system of the first order conditions when success is private are (weakly) greater than the coefficients corresponding to the (𝐿𝐿𝑆 ) for 𝑡 ≤ 𝑇 and (𝐿𝐿𝐹 ) for a system of the first order conditions when success is public.

Proof: Recall that the Kuhn-Tucker conditions when success was public were as follows:

ℒ

= 𝑃 𝑓 (𝑡) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (𝑡) + .

We will show that if we solve the system of equations recursively coefficients 𝛼 are (weakly) greater than those that solve the Kuhn Tucker conditions for the optimization problem when success is public. More precisely, consider

= 0:

𝜕ℒ

𝜕𝑦 = 𝛽 𝛿

𝜓 𝑃 𝑓 (1) 𝜐𝑃 + (1 − 𝜐)𝑃 − 𝜉

𝛿 + 𝜉

𝛿 𝑃 𝑓 (1) + 𝜓

𝛽 𝛿 𝛼 1{𝑡 < 𝑇 }

−𝛿𝛼 1{𝑡 > 1}

+𝛼 1{𝑡 = 𝑇 } 1 − =

𝑃 𝑓 (1) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (1) + 𝛼 = 0.

As you can see, this is exactly the same equation as

= 0 for the system with public success.

Next, for private success, consider

= 0:

ℒ

= 𝑃 𝑓 (2) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (2) + 𝛼 1{𝑡 < 𝑇 }

−𝛿𝛼 1{𝑡 > 1}

+𝛼 1{𝑡 = 𝑇 } 1 − =

𝑃 𝑓 (2) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (2) + 𝛼 − 𝛿𝛼 = 0.

At the same time, the corresponding condition

= 0 for the system with public success is

3

See Appendix A, Case B for more details.

14

𝑃 𝑓 (2) 𝜐𝑃 + (1 − 𝜐)𝑃 − + 𝑃 𝑓 (2) + 𝛼 = 0,

which immediately implies that 𝛼 corresponding to private success is (weakly) greater than 𝛼 corresponding to public success.

Repeating this procedure for 𝑡 = 3, … 𝑇 the statement for 𝛼 is proven.

Q.E.D.

The result of Lemma C.1 implies that if a limited liability constraint is binding when success is public then the corresponding ex post moral hazard constraint must be binding as well.

The implication is that, when the agent must be paid on top of what is required by the ex post moral hazard constraints, he must be paid this additional amount in the same periods as in Case B of Appendix A.

Finally, Lagrange multipliers 𝜉 , 𝜉 , {𝛼 } , {𝛼 } for a system of the first order conditions when success is private are those 𝜉 , 𝜉 , {𝛼 } , {𝛼 } for a system of the first order conditions when success is public, except that {𝛼 } , {𝛼 } when success is private are (weakly) greater than the corresponding coefficients when success is public.

Recall that when success was public, there were two sets of Lagrange multipliers. The same will be true when success is private. First, 𝜉 > 0 and 𝜉 > 0 ⟺ 𝑇 > 𝑇

, 𝛼 > 0 for 𝑡 <

𝑇 , 𝛼 = 0 and 𝛼 > 0 for 𝑡 > 1, 𝛼 = 0. Since in our case the Lagrange multipliers are weakly greater, we have 𝜉 > 0 and 𝜉 > 0. Then 𝛼 > 0 for 𝑡 < 𝑇 , 𝛼 = 0 and 𝛼 > 0 for 𝑡 > 1, 𝛼 = 0 . Second, 𝜉 = 0 and 𝜉 > 0 ⟺ 𝑇 ≤ 𝑇 , 𝛼 > 0 for 𝑡 ≤ 𝑇 . This implies when success is private we have 𝜉 = 0 and 𝜉 > 0, 𝛼 > 0 for 𝑡 ≤ 𝑇 and 𝛼 > 0 for 𝑡 > 1, 𝛼 = 0. Thus, distortions in the output and duration of the experimentation stages when success is private are of the same nature as in Case B of Appendix A.

We now explicitly derive the optimal payment scheme for each type. For the high type, it is optimal to set 𝑥 = 0, 𝑦 = 0 for 1 < 𝑡 ≤ 𝑇 and 𝑦 > 0, where the exact value of 𝑦 is determined by the binding (𝐼𝐶

) and (𝐼𝐶

) constraints. For the low type, the exact payment scheme depends on the value of 𝑇 . If 𝑇 ≤ 𝑇 it is optimal to set 𝑥 > 0 and 𝑦 =

max 𝛿 𝑥 + 𝛿 𝑐 − 𝑐 𝑞 𝑐 , 0 for 𝑡 ≤ 𝑇 . If 𝑇 > 𝑇 then it is optimal to set 𝑥 = 0, 𝑦 = 𝛿 𝑦 for 𝑡 < 𝑇 , and 𝑦 = 𝑐 − 𝑐 𝑞 𝑐 + 𝜏 > 0, where the exact value of 𝜏 is determined by the binding (𝐼𝐶

) and (𝐼𝐶

) constraints

As for the high type, the exact value of 𝑥 and 𝑦 for each case is determined by the binding (𝐼𝐶

) and (𝐼𝐶

) constraints.

To derive the exact values of 𝑦 , 𝑥 , and 𝑦 , we consider each case (𝑇 ≤ 𝑇 and 𝑇 >

𝑇 ) in turn below.

15 𝑻

≤ 𝑻

: If 𝑇 ≤ 𝑇 the (𝐼𝐶) constraints become:

(𝐼𝐶

)

𝛽 𝛿𝜆 𝑦 = 𝛽 𝛿 𝜆 ∑ (1 − 𝜆 ) max 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 , 0 +

𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝛿 𝑃 𝑥 ,

(𝐼𝐶

) 𝛽 𝛿 ∑ (1 − 𝜆 ) 𝜆 max 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 , 0 + 𝛿 𝑃 𝑥 =

𝛽 𝛿𝜆 𝑦 + 𝛿 𝑃 𝑥 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

There are two possible solutions depending on the value of 𝑥 .

First, if 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 ≤ 0, then the (𝐼𝐶) constraints could be rewritten as (𝐼𝐶

) 𝛽 𝛿𝜆 𝑦 = 𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝛿 𝑃 𝑥 ,

(𝐼𝐶

) 𝛿 𝑃 𝑥 = 𝛽 𝛿𝜆 𝑦 + 𝛿 𝑃 𝑥 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

Solving for 𝑥 and 𝑦 we have:

𝑥 =

, and

𝑦 =

=

∆

∆ ∆

=

∆ ∆

.

Second, if 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 > 0, then the (𝐼𝐶) constraints could be rewritten as (𝐼𝐶

)

𝛽 𝛿𝜆 𝑦 = 𝛽 𝛿 1 − (1 − 𝜆 ) 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 +

𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝛿 𝑃 𝑥 ,

(𝐼𝐶

) 𝛽 𝛿 1 − (1 − 𝜆 ) 𝑥 + 𝑐 − 𝑐 𝑞 𝑐 + 𝛿 𝑃 𝑥 =

16

𝛽 𝛿𝜆 𝑦 + 𝛿 𝑃 𝑥 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

Solving for 𝑥 and 𝑦 we have:

𝑥 =

⎝

⎜

⎛

𝛽 𝑐 − 𝑐 𝑞 𝑐 𝜆 1 − (1 − 𝜆 ) − 𝜆 1 − (1 − 𝜆 )

+𝛽 𝛿 𝑐 − 𝑐 𝑞 𝑐 𝜆 1 − (1 − 𝜆 ) − 𝜆 1 − (1 − 𝜆 )

−𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝑃 ∆𝑐 𝑞 𝑐 ⎠

⎟

⎞ ,

𝑦 =

⎝

⎛

𝛽 𝛿 𝑐 − 𝑐 𝑞 𝑐 (1 − 𝜆 ) − (1 − 𝜆 )

+𝛽 𝛿 𝑐 − 𝑐 𝑞 𝑐 (1 − 𝜆 ) − (1 − 𝜆 )

−𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝛿 𝑃 ∆𝑐 𝑞 𝑐 ⎠

⎞.

𝑻

> 𝑻

. If 𝑇 > 𝑇 the (𝐼𝐶) constraints become:

(𝐼𝐶

) 𝛽 𝛿𝜆 𝑦 = 𝛽 𝛿 𝑐 − 𝑐 𝑞 𝑐 + 𝜏 1 − (1 − 𝜆 ) +

𝛿 𝑃 ∆𝑐 𝑞 𝑐 ,

(𝐼𝐶

) 𝛽 𝛿 𝑐 − 𝑐 𝑞 𝑐 + 𝜏 1 − (1 − 𝜆 )

= 𝛽 𝛿𝜆 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

Solving for 𝑦 and 𝑦 we have:

𝜏

=

⎜

⎛ ₊₁ 𝑐

+1

∆ +1 +1 ∆

+1 +1 ⎠

⎟

⎞

− 𝑐

𝑇^𝐿+1

𝐿

− 𝑐 𝑞

𝑐

𝑇^𝐿+1

𝐿

;

𝑦₁^𝐻

=

1

𝛽₀𝛿 𝜆^𝐻 1− 1−𝜆^{𝐿 𝑇} 𝐿

−𝜆^𝐿 1− 1−𝜆^{𝐻 𝑇} 𝐿

⎝

⎜ ⎜

⎛ ^𝛽

^𝛿

𝑇^𝐻

𝑐

𝐻

− 𝑐 𝑞

𝑐

𝐻

1 − ( 1 − 𝜆

)

1 − ( 1 − 𝜆

)

− 1 − ( 1 − 𝜆

)

1 − ( 1 − 𝜆

)

+𝛿

𝑃

∆𝑐

𝑞

𝑐

𝐿

1 − ( 1 − 𝜆

)

−𝛿

𝑃

∆𝑐

𝑞

𝑐

𝐻

1 − ( 1 − 𝜆

)

⎠

⎟ ⎟

⎞ ,

and 𝑦 = 𝑐 − 𝑐 𝑞 𝑐 + 𝜏.

17

Case 2: The (𝑰𝑪

) constraint binds and (𝑰𝑪

) constraint is slack.

Lemma B.2: 𝜓 > 0 and 𝜓 = 0 ⟹ 𝜉 , {𝛼 } , 𝛼 > 0 (it is optimal to set 𝑥 = 0, 𝑦 = 0 for 𝑡 ≤ 𝑇 ).

Proof: We can rewrite the Kuhn-Tucker conditions as follows:

ℒ

= −(1 − 𝜐)𝛿 𝑃 − 𝜓 𝛿 𝑃 + 𝜉 − 𝛼 ⟹ 𝜉 = (1 − 𝜐)𝛿 𝑃 + 𝛼 + 𝜓 𝛿 𝑃 > 0.

Solving recursively, we have

𝛼 = (1 − 𝜐)𝛽 𝛿 (1 − (1 − 𝜆 ) ) + 𝜓 𝛽 𝛿 (1 − (1 − 𝜆 ) ) > 0 for 𝑡 < 𝑇 and 𝛼 = (1 − 𝜐)𝛽 𝛿 1 − (1 − 𝜆 ) + 𝜓 𝛽 𝛿 1 − (1 − 𝜆 ) > 0. Thus, 𝜉 = 𝛿 (1 − 𝜐) 𝑃 + 𝛽 1 − (1 − 𝜆 ) + 𝜓 𝑃 + 𝛽 1 − (1 − 𝜆 ) > 0.

Q.E.D.

In this case the (𝐼𝐶) constraints become:

(𝐼𝐶

) 𝛽 𝛿𝜆 𝑦 = 𝛿 𝑃 ∆𝑐 𝑞 𝑐 , (𝐼𝐶

) 0 > 𝛽 𝛿𝜆 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

II. Optimal length of experimentation Case 1 [both the (𝐼𝐶

) and (𝐼𝐶

) constraints bind]

Since some of the Lagrange multipliers are now positive, we can have distortion in both 𝑇 and 𝑇 as determined by

= 0 and

= 0. It is possible, in general, to have 𝑇 > 𝑇 or 𝑇 < 𝑇 and 𝑇 > 𝑇 or 𝑇 < 𝑇 .

Case 2 [the (𝐼𝐶

) constraint binds and (𝐼𝐶

) constraint is slack]

In this case, we can now have distortion in both 𝑇 and 𝑇 . In particular, the F.O.C. with respect to 𝑇 becomes:

ℒ

=

= 0.

Since the informational rent of the high-type agent is non-monotonic in 𝑇 , it is possible, in general, to have 𝑇 > 𝑇 or 𝑇 < 𝑇 .

Also, there will be a distortion in the duration of the experimentation stage for the high type:

ℒ

= = 0,

18

or, equivalently, 𝑇 = 𝑇 when 𝛼 = 0. In case 𝛼 > 0, we have = 𝛼 𝑞 𝑐 . This, in turn, implies that 𝑇 < 𝑇 as > 0.

III. Optimal outputs Case 1 [both the (𝐼𝐶

) and (𝐼𝐶

) constraints bind]

To characterize the optimal output choices, we now consider the following Kuhn-Tucker conditions for the optimization problem:

[𝑞 (𝑐)] 𝜐𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 (𝑐) − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ; [𝑞 (𝑐)] (1 − 𝜐)𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 (𝑐) − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ;

𝑞 𝑐

= − 𝛼 𝑐 − 𝛼 = 0;

𝑞 𝑐

= 0 = 0.

The first two conditions above imply that there is no distortion in the output relative to the first-best level after success has been observed by either type; that is 𝑉 𝑞 (𝑐) = 𝑐 for 𝜃 ∈ {𝐻, 𝐿}. There is under production for the high type: 𝑞 𝑐 < 𝑞 𝑐 . It is possible, in general, to have 𝑞 𝑐 > 𝑞 𝑐 or 𝑞 𝑐 < 𝑞 𝑐

Case 2 [the (𝐼𝐶

) constraint binds and (𝐼𝐶

) constraint is slack]

To characterize the optimal output choices, we now consider the following Kuhn-Tucker conditions for the optimization problem:

𝑞 (𝑐) 𝜐𝛽 𝛿 1 − 𝜆 𝜆 𝑉 𝑞 (𝑐) − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ;

𝑞 𝑐 𝜐𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 − 𝛼 𝑐 − 𝑐 = 0;

𝑞 𝑐 (1 − 𝜐)𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 + 𝜓 𝛿 𝑃 ∆𝑐

−𝛼 𝑐 − 𝑐 = 0.

The first condition above implies that there is no distortion in the output relative to the first-best level after success has been observed by either type; that is 𝑉 𝑞 (𝑐) = 𝑐 for 𝜃 ∈ {𝐻, 𝐿}.

However, there is distortion in the output relative to the first-best level after 𝑇 failures have been reported by the high type; that is

𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 = −𝜓 𝛿 𝑃 ∆𝑐 + (1 − 𝜐)𝛽 𝛿 1 −

(1 − 𝜆 ) + 𝛼 𝛽 𝛿 1 − (1 − 𝜆 ) 𝑐 − 𝑐 = 0,

which implies that 𝑞 𝑐 < 𝑞 𝑐 .

19

For the high type, there is under production when 𝛼 > 0:

𝜐𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 − 𝛽 𝛿 1 − (1 − 𝜆 ) (𝜐 − 𝜓 ) 𝑐 − 𝑐 = 0 or,

equivalently, 𝑞 𝑐 < 𝑞 𝑐 since 𝜓 ≤ 𝜐.

Appendix D (Learning good news)

We summarize the main results as Proposition 4.

Proposition 4

i) There are two cases to consider.

Case A: Only the low type’s IC is binding.

There is no restriction on when to reward the high type.

Case B: Both types’ IC are binding.

If the cost of experimentation is large ( 𝛾 > 𝛾

), the principal must reward the low type after failure and the high-type for early success (in the very first period). If the cost of experimentation is small (𝛾 < 𝛾

) , the principal must reward the low type after late success (last period) and the high-type for early success (in the very first period) ii) In the optimal contract, each type may over experiment relative to the first best.

iii) After failure, the high type under-produces relative to the first best output. The low type over-produces if the high type receives a rent and produces at the first best level otherwise. After success, each type produces at the first best level.

The proof of Proposition 4 mirrors the proof of the bad news case (See Appendix A).

The key difference is that now rewards and the output after success are functions of 𝑐: 𝑦 ≡ 𝑤 𝑐 − 𝑐𝑞 𝑐 . Also recall that now the difference in the expected cost, ∆𝑐 = 𝑐 − 𝑐 , is negative since the 𝐻 type is relatively more pessimistic after the same amount of failures that is, 𝑐 > 𝑐 .

We first characterize the optimal payment structure, 𝑥 , {𝑦 } , 𝑥 and {𝑦 } (part (i) of Proposition 4), then the optimal length of experimentation, 𝑇 and 𝑇 (part (ii) of Proposition 4), and finally the optimal outputs 𝑞 𝑐 , 𝑞 𝑐 , 𝑞 𝑐 and 𝑞 𝑐 (part (iii) of Proposition 4).

Denote the expected surplus net of costs for 𝜃 = 𝐻, 𝐿 by Ω 𝜛 = 𝛽 ∑ 𝛿 1 −

𝜆 𝜆 𝑉 𝑞 𝑐 − 𝑐𝑞 𝑐 − 𝛤 + 𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 𝑞 𝑐 − 𝛤 and

20

define 𝑦 as the rent payment to the 𝜃 type who succeeds in period 𝑡 and 𝑥 as the rent payment to the 𝜃 type who failed during the entire experimentation stage: 𝑦 ≡ 𝑤 𝑐 − 𝑐𝑞 𝑐 for 1 ≤

𝑡 ≤ 𝑇 , and 𝑥 ≡ 𝑤 𝑐 − 𝑐 𝑞 𝑐 .

The principal’s optimization problem then is to choose contracts 𝜛 and 𝜛 to maximize the expected net surplus minus rent of the agent, subject to the respective 𝐼𝐶 and 𝐿𝐿 constraints given below:

𝑀𝑎𝑥 𝐸 Ω 𝜛 − 𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑦 − 𝛿 𝑃 𝑥 subject to:

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥

≥ 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 + ∆𝑐 𝑞 𝑐 ,

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥

≥ 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 − ∆𝑐 𝑞 𝑐 ,

𝐿𝐿𝑆 𝑦 ≥ 0 for 𝑡 ≤ 𝑇 , 𝐿𝐿𝐹 𝑥 ≥ 0 for 𝜃 = 𝐻, 𝐿.

We begin to solve the problem by first proving the following claim.

Claim: The constraint (𝐼𝐶

) is binding and the low type obtains a strictly positive rent.

Proof: If the (𝐼𝐶

) constraint was not binding, it would be possible to decrease the payment to the low type until (𝐿𝐿𝑆 ) and (𝐿𝐿𝐹 ) are binding, but that would violate (𝐼𝐶

) since

−∆𝑐 𝑞 𝑐 > 0. Q.E.D.

I. Optimal payment structure, 𝒙

, {𝒚

}

, 𝒙

and {𝒚

}

(part (i) of Proposition 4)

We will show that the solution to the principal’s optimization problem depends on whether the (𝐼𝐶

) constraint is binding or not; we explore each case separately in what follows.

Case A: The (𝑰𝑪

) constraint is not binding.

In this case the high type does not receive any rent and it immediately follows that 𝑥 = 0 and 𝑦 = 0 for 1 ≤ 𝑡 ≤ 𝑇 , which implies that the rent of the low type in this case becomes

−𝛿 𝑃 ∆𝑐 𝑞 𝑐 . Replacing 𝑥 in the objective function, the principal’s

21

optimization problem is to choose 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 , 𝑇 , {𝑦 } , 𝑞 𝑐 and

𝑞 𝑐 to

𝑀𝑎𝑥 𝐸 𝜋 𝜛 + (1 − 𝜐)𝛿 𝑃 ∆𝑐 𝑞 𝑐 subject to:

(𝐿𝐿𝑆 ) 𝑦 ≥ 0 for 𝑡 ≤ 𝑇 ,

and (𝐿𝐿𝐹 ) −𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 ≥ 0.

When the (𝐼𝐶

) constraint is not binding, the claim below shows that there are no restrictions in choosing {𝑦 } except those imposed by the (𝐼𝐶

) constraint. In other words, the principal can choose any combinations of nonnegative payments to the low type

𝑥 , {𝑦 } such that 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 =

−𝛿 𝑃 ∆𝑐 𝑞 𝑐 . Labeling by {𝛼 } , 𝛼 the Lagrange multipliers of the constraints associated with (𝐿𝐿𝑆 ) for 𝑡 ≤ 𝑇 , and (𝐿𝐿𝐹 ) respectively, we have the following claim.

Claim A.1: If (𝐼𝐶

) is not binding, we have 𝛼 = 0 and 𝛼 = 0 for all 𝑡 ≤ 𝑇 . Proof: We can rewrite the Kuhn-Tucker conditions as follows:

ℒ

= 𝛼 − 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ;

ℒ

= 𝑦 ≥ 0; 𝛼 ≥ 0; 𝛼 𝑦 = 0 for 1 ≤ 𝑡 ≤ 𝑇 . Suppose to the contrary that 𝛼 > 0. Then,

−𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 = 0,

and there must exist 𝑦 > 0 for some 1 ≤ 𝑠 ≤ 𝑇 . Then, we have 𝛼 = 0, which leads to a contradiction since

= 0 cannot be satisfied unless 𝛼 = 0.

Suppose to the contrary that 𝛼 > 0 for some 1 ≤ 𝑠 ≤ 𝑇 . Then, 𝛼 > 0, which leads to a

contradiction as we have just shown above. Q.E.D.

Case B: The (𝑰𝑪

) constraint is binding.

When both (𝐼𝐶) constraints are binding, the optimal payment structure resembles the case of learning bad news. Denoting by 𝜓 = 𝑃 𝑃 − 𝑃 𝑃 , we can re-write the incentive compatibility constraints as:

𝑥 𝛿 𝜓 = 𝛽 ∑ 𝛿 𝑃 (1 − 𝜆 ) 𝜆 − 𝑃 (1 − 𝜆 ) 𝜆 𝑦 +𝛽 ∑ 𝛿 𝑃 (1 − 𝜆 ) 𝜆 − 𝑃 (1 − 𝜆 ) 𝜆 𝑦

+𝑃 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 , and

𝑥 𝛿 𝜓 = 𝛽 ∑ 𝛿 𝑃 (1 − 𝜆 ) 𝜆 − 𝑃 (1 − 𝜆 ) 𝜆 𝑦

22

+𝛽 ∑ 𝛿 𝑃 (1 − 𝜆 ) 𝜆 − 𝑃 (1 − 𝜆 ) 𝜆 𝑦

+𝑃 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

Case B.1: 𝜓 = 𝑃 𝑃 − 𝑃 𝑃 ≠ 0.

First, we consider the case when 𝜓 ≠ 0. This is when the likelihood ratio of reaching the last period of the experimentation stage is different for both types i.e., when ≠ . In

Lemma D.3, we establish that the high type is not rewarded for failure. In Lemma D.4, we prove that the low type is rewarded for failure if and only if 𝑇 ≤ 𝑇 and, in Lemma D.5, that he is rewarded for the very last success if 𝑇 > 𝑇 . We also show that the high type may be rewarded only for the very first success.

Then 𝑥 and 𝑥 can be expressed as functions of {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 and 𝑞 𝑐 only from the binding (𝐼𝐶

) and (𝐼𝐶

). The principal’s optimization problem is to choose 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 , {𝑦 } , 𝑇 , {𝑦 } , 𝑞 𝑐 , and 𝑞 𝑐 to

𝑀𝑎𝑥 𝐸 Ω 𝜛 − 𝛿 𝑃 𝑥 {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐

−𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑦

subject to

𝐿𝐿𝑆 𝑦 ≥ 0 for 𝑡 ≤ 𝑇 ,

𝐿𝐿𝐹 𝑥 {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 ≥ 0 for 𝜃 = 𝐻, 𝐿.

Labeling {𝛼 } , {𝛼 } , 𝜉 and 𝜉 as the Lagrange multipliers of the constraints associated with (𝐿𝐿𝑆 ), (𝐿𝐿𝑆 ), (𝐿𝐿𝐹 ) and (𝐿𝐿𝐹 ) respectively, the Lagrangian is:

ℒ = 𝐸 Ω 𝜛 − 𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑦 −

𝛿 𝑃 𝑥 {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐

+ 𝛼 𝑦 + 𝛼 𝑦 + 𝜉 𝑥 {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 +𝜉 𝑥 {𝑦 } , {𝑦 } , 𝑇 , 𝑇 , 𝑞 𝑐 , 𝑞 𝑐 .

The Inada conditions give us interior solutions for 𝑞 𝑐 , 𝑞 𝑐 , 𝑞 𝑐 and 𝑞 𝑐 . We also assumed that 𝑇 > 0 and 𝑇 > 0. The Kuhn-Tucker conditions are the same as in case of learning good news; we refer to Appendix A (Case B.1) for the details.

Lemma C.3: The high type does not get rent after failure, i.e., 𝑥 = 0.

Proof: See Lemma 3 in Appendix A.

23

Lemma C.4. 𝜉 = 0 ⇒ 𝑇 ≤ 𝑇 , 𝛼 > 0 for 𝑡 ≤ 𝑇 (it is optimal to set 𝑥 > 0, 𝑦 = 0 for 𝑡 ≤ 𝑇 ) and 𝛼 > 0 for all 𝑡 > 1 and 𝛼 = 0 (it is optimal to set 𝑥 = 0, 𝑦 = 0 for all 𝑡 > 1 and 𝑦 > 0).

Proof: See Lemma 4 in Appendix A.

Lemma C.5: 𝜉 > 0 ⇒ 𝑇 > 𝑇 , 𝛼 > 0 for 𝑡 < 𝑇 , 𝛼 = 0 and 𝛼 > 0 for 𝑡 > 1, 𝛼 = 0 (it is optimal to set 𝑥 = 0, 𝑦 = 0 for 𝑡 < 𝑇 , 𝑦 > 0 and 𝑥 = 0, 𝑦 = 0 for 𝑡 > 1, 𝑦 > 0) Proof: See Lemma 5 in Appendix A.

Case B.2: 𝜓 = 𝑃 𝑃 − 𝑃 𝑃 = 0.

Finally, we analyze the case when = . In this case, the likelihood ratio of reaching the last period of the experimentation stage is the same for both types and 𝑥 and 𝑥 cannot be used as screening variables. Therefore, the principal must reward both types for success and she chooses 𝑇 > 𝑇 .

Lemma C.2.1. 𝑃 𝑃 − 𝑃 𝑃 = 0 ⇒ 𝑇 > 𝑇 , 𝜉 > 0, 𝜉 > 0, 𝛼 > 0 for 𝑡 > 1 and 𝛼 >

0 for 𝑡 < 𝑇 (it is optimal to set 𝑥 = 𝑥 = 0, 𝑦 = 0 for 𝑡 > 1 and 𝑦 = 0 for 𝑡 < 𝑇 ).

Proof: See Lemma B.2.1 in Appendix A.

II. Optimal length of experimentation (part (ii) of Proposition 4)

Since 𝑇 and 𝑇 affect the information rents, 𝑈 and 𝑈 , there will be a distortion in the duration of the experimentation stage for both types:

ℒ

=

= 0.

The exact values of 𝑈 and 𝑈 depend on whether we are in Case A ((𝐼𝐶

) is slack) or Case B (both (𝐼𝐶

) and (𝐼𝐶

) are binding.)

In Case A, by Claim A.1, the low type’s rent −𝛿 𝑃 ∆𝑐 𝑞 𝑐 is not affected by 𝑇 . Therefore, the F.O.C. with respect to 𝑇 is identical to that under first best:

ℒ

= = 0,

or, equivalently, 𝑇 = 𝑇 when (𝐼𝐶

) is not binding. However, since the low type’s

information rent depends on 𝑇 , there will be a distortion in the duration of the experimentation stage for the high type:

ℒ

=

( ) ∆

= 0.

24

Since the informational rent of the low-type agent, −𝛿 𝑃 ∆𝑐 𝑞 𝑐 , is non- monotonic in 𝑇 , it is possible, in general, to have 𝑇 > 𝑇 or 𝑇 < 𝑇 .

In Case B, the exact values of 𝑈 and 𝑈 depend on whether 𝑇 < 𝑇 (Lemma D.4) or 𝑇 > 𝑇 (Lemma D.5), but in each case 𝑈 > 0 and 𝑈 ≥ 0. It is possible, in general, to have 𝑇 > 𝑇 or 𝑇 < 𝑇 and 𝑇 > 𝑇 or 𝑇 < 𝑇 .

III. Optimal outputs (part (iii) of Proposition 4)

The next two conditions imply that there is no distortion in the output relative to the first-best level after success has been observed by either type:

𝜐𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 𝑐 − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ; (1 − 𝜐)𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 𝑐 − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 . Case A [when (𝐼𝐶

) is not binding]

The following two FOCs imply that there is no distortion after failure by the low type but there will be underproduction by the high type after failure, that is, 𝑞 𝑐 < 𝑞 𝑐 :

𝑉 𝑞 𝑐 − 𝑐 = −

∆𝑐 ,

𝑉 𝑞 𝑐 − 𝑐 = 0.

Case B. [when (𝐼𝐶

) is binding]

The following two FOCs imply that there will be overproduction for the low type (𝑞 𝑐 >

𝑞 𝑐 ) and underproduction for the high type (𝑞 𝑐 < 𝑞 𝑐 ) after failure.

We start with the main case B.1, when 𝜓 ≠ 0, and consider cases when 𝑇 ≤ 𝑇 and 𝑇 > 𝑇 separately.

When 𝑇 ≤ 𝑇 , we have:

(1 − 𝜐) 𝑉 𝑞 𝑐 − 𝑐 =

,

𝜐𝑃 𝑉 𝑞 𝑐 − 𝑐 = −

.

When 𝑇 > 𝑇 , we have:

𝑉 𝑞 𝑐 − 𝑐 =

( ) ( ) ( )

∆𝑐 ,

𝑉 𝑞 𝑐 − 𝑐 = −

( ) ( )

∆𝑐 ,

In the knife-edge case B.2, when 𝜓 = 0, the relevant FOCs are:

25

𝑉 𝑞 𝑐 − 𝑐 = −

( , )

,

𝑉 𝑞 𝑐 − 𝑐 =

( , )

.

Q.E.D.

Appendix E (Identical 𝑇)

Suppose the principal must choose an identical length of the experimentation stage for both types, 𝑇 = 𝑇 . We prove that (𝐼𝐶

) is not binding in this case. A contract is now defined formally by 𝜛 = 𝑇, 𝑤 (𝑐), 𝑞 (𝑐) , 𝑤 𝑐 , 𝑞 𝑐 , where 𝑇 is the maximum duration of the experimentation stage regardless of the announced type.

1. The first best

We present the first best case when the agent’s type 𝜃 is common knowledge before the principal offers the contract. The principal’s objective function is:

𝜐𝜋 (𝜛 ) + (1 − 𝜐)𝜋 (𝜛 ) =

𝜐 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) − 𝛤

+𝛿 1 − 𝛽 + 𝛽 (1 − 𝜆 ) 𝑉 𝑞 𝑐 − 𝑐 𝑞 𝑐 − 𝛤

+(1 − 𝜐) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) − 𝛤

+𝛿 1 − 𝛽 + 𝛽 (1 − 𝜆 ) 𝑉 𝑞 𝑐 − 𝑐 𝑞 𝑐 − 𝛤

.

Since the expected cost is rising as long as success is not obtained for both types, the first-best solution is characterized by a termination date 𝑇 .

The optimal 𝑇 is bounded and it is the highest 𝑡 such that

𝜐 𝛿𝛽 𝜆 𝑉 𝑞 ( 𝑐 ) − 𝑐 𝑞 ( 𝑐 ) + 𝛿(1 − 𝛽 𝜆 ) 𝑉 𝑞 (𝑐 ) − 𝑐 𝑞 (𝑐 ) +(1 − 𝜐) 𝛿𝛽 𝜆 𝑉 𝑞 ( 𝑐 ) − 𝑐 𝑞 ( 𝑐 ) + 𝛿(1 − 𝛽 𝜆 ) 𝑉 𝑞 (𝑐 ) − 𝑐 𝑞 (𝑐 )

≥ 𝛾 + 𝜐 𝑉 𝑞 (𝑐 ) − 𝑐 𝑞 (𝑐 ) + (1 − 𝜐) 𝑉 𝑞 (𝑐 ) − 𝑐 𝑞 (𝑐 )

26

2. Asymmetric information

We first characterize the optimal payment structure, 𝑥 , {𝑦 }

, 𝑥 and {𝑦 }

, then the optimal length of experimentation, 𝑇 , and finally the optimal outputs {𝑞 ( 𝑐 )}

, 𝑞 𝑐

, {𝑞 ( 𝑐 )}

and 𝑞 𝑐

.

Denote the expected surplus net of costs for 𝜃 = 𝐻, 𝐿 by Ω 𝜛 = 𝛽 ∑ 𝛿 1 −

𝜆 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) − 𝛤 + 𝛿 𝑃 𝑉 𝑞 𝑐 − 𝑐 𝑞 𝑐 − 𝛤 . The

principal’s optimization problem then is to choose contracts 𝜛 and 𝜛 to maximize the expected net surplus minus rent of the agent, subject to the respective 𝐼𝐶 and 𝐿𝐿 constraints given below:

𝑀𝑎𝑥 𝐸 Ω 𝜛 − 𝛽 ∑ 𝛿 1 − 𝜆 𝜆 𝑦 − 𝛿 𝑃 𝑥 subject to:

(𝐼𝐶

) 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 (𝑦 − 𝑦 ) + 𝛿

𝑃

𝑥 − 𝑥 − ∆𝑐

𝑞 𝑐

≥ 0,

(𝐼𝐶

) 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 (𝑦 − 𝑦 ) + 𝛿

𝑃

𝑥 − 𝑥 + ∆𝑐

𝑞 𝑐

≥ 0, 𝐿𝐿𝑆 𝑦 ≥ 0 for 𝑡 ≤ 𝑇 ,

𝐿𝐿𝐹

𝑥 ≥ 0.

We begin to solve the problem by first proving the following claim.

Claim: The constraint (𝐼𝐶

) is binding and the high type obtains a strictly positive rent.

Proof: If the (𝐼𝐶

) constraint was not binding, it would be possible to decrease the payment to the high type until (𝐿𝐿𝑆 ) and (𝐿𝐿𝐹 ) are binding, but that would violate (𝐼𝐶

) since

∆𝑐

𝑞 𝑐

> 0. Q.E.D.

I. Optimal payment structure, 𝒙

, {𝒚

}

, 𝒙

and {𝒚

}

We will now show that the (𝐼𝐶

) constraint is not binding in this problem and that the low type gets zero rent.

Lemma D: The (𝐼𝐶

) is not binding.

Proof: Labeling 𝛼 , 𝛼 , {𝛼 } , {𝛼 } , 𝜉 and 𝜉 as the Lagrange multipliers of the constraints associated with (𝐼𝐶

), (𝐼𝐶

), (𝐿𝐿𝑆 ), (𝐿𝐿𝑆 ), (𝐿𝐿𝐹 ) and (𝐿𝐿𝐹 ) respectively, we have the following Kuhn-Tucker conditions for the optimization problem:

ℒ

= −𝜐𝛽 𝛿 (1 − 𝜆 ) 𝜆 + 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 − 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 = −𝛼 ;

ℒ

= −(1 − 𝜐)𝛽 𝛿 (1 − 𝜆 ) 𝜆 − 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 + 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 = −𝛼 ;

27

ℒ

= −𝜐𝛿 𝑃 + 𝛼 𝛿 𝑃 − 𝛼 𝛿 𝑃 + 𝜉 = 0;

ℒ

= −(1 − 𝜐)𝛿 𝑃 − 𝛼 𝛿 𝑃 + 𝛼 𝛿 𝑃 + 𝜉 = 0.

After summing up

= 0 and

= 0 for 1 ≤ 𝑡 ≤ 𝑇 we have

𝛼 + 𝛼 = 𝛽 𝛿 (𝜐(1 − 𝜆 ) 𝜆 + (1 − 𝜐)(1 − 𝜆 ) 𝜆 ).

After summing up

= 0 and

= 0 we have 𝜉 + 𝜉 = 𝛿 𝐸 𝑃 .

From

= 0 we express 𝛼 = 𝛼 − + 𝜐, and plug it into

= 0 to derive 𝛼 = 𝛼 𝛽 𝛿 𝑓 𝑡, 𝑇 + 𝛽 𝛿 (1 − 𝜆 ) 𝜆 for 𝑡 ≤ 𝑇, where is defined similarly to 𝑓 (𝑡, 𝑇 ):

𝑓 𝑡, 𝑇 = (1 − 𝜆 ) 𝜆 − (1 − 𝜆 ) 𝜆 .

Lemma D.1: There exists a unique 𝑇

> 1, such that 𝑓 𝑇

, 𝑇 = 0, and 𝑓 𝑡, 𝑇 < 0 for 𝑡 < 𝑇

> 0 for 𝑡 > 𝑇

.

Proof: Follows from Lemma 1 in Appendix A (replace 𝑇 with 𝑇.) Q.E.D.

From

= 0 we express 𝛼 = 𝛼 − + (1 − 𝜐) and plug it into

= 0 to derive 𝛼 = −𝛼 𝛽 𝛿 𝑓 𝑡, 𝑇 + 𝛽 𝛿 (1 − 𝜆 ) 𝜆 for 𝑡 ≤ 𝑇.

Consider 𝛼 = 𝛼 𝛽 𝛿 𝑓 𝑡, 𝑇 + 𝛽 𝛿 (1 − 𝜆 ) 𝜆 for 𝑡 ≤ 𝑇:

Since 𝑓 𝑡, 𝑇 < 0 for 𝑡 < 𝑇

𝑇 we immediately conclude that 𝜉 > 0. In addition, for 𝑡 >

𝑇

𝑇 it must be that 𝛼 > 0.

Consider 𝛼 = −𝛼 𝛽 𝛿 𝑓 𝑡, 𝑇 + 𝛽 𝛿 (1 − 𝜆 ) 𝜆 for 𝑡 ≤ 𝑇:

Since 𝑓 𝑡, 𝑇 < 0 for 𝑡 < 𝑇

𝑇 we immediately conclude that 𝛼 > 0 for 𝑡 < 𝑇

𝑇 . In addition, if 𝑇 > 𝑇

𝑇 it must be that 𝜉 > 0.

We now prove that (𝐼𝐶

) is not binding by contradiction. We consider two cases, 𝑇 <

𝑇

𝑇 and 𝑇 ≥ 𝑇

𝑇 , in turn.

𝑻 < 𝑻

𝑻 . Suppose there exists a solution with 𝑇 < 𝑇

𝑇 . Then it must be that 𝑥 > 0, 𝑦 = 0 for 𝑡 ≤ 𝑇 and 𝑥 = 0. The incentive compatibility constraints might be rewritten as

(𝐼𝐶

) 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 + 𝑥 = 0,

(𝐼𝐶

) 𝑥 =

− ∆𝑐 𝑞 𝑐 .

Combining the two equations together we have

28

𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 −

𝛿 𝑃 ∆𝑐 𝑞 𝑐 +

− ∆𝑐 𝑞 𝑐 = 0,

∑ 𝛿 (1 − 𝜆 ) 𝜆 − (1 − 𝜆 ) 𝜆 𝑦 + 𝑃 ∆𝑐 𝑞 𝑐 − 𝑞 𝑐 = 0,

∑ 𝛿 𝑓 𝑡, 𝑇 𝑦 + 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝑞 𝑐 = 0 and we a contradiction.

𝑻 ≥ 𝑻

𝑻 . Suppose there exists a solution with 𝑇 > 𝑇

𝑇 . Then it must be that 𝑥 = 0, 𝑦 = 0 for 𝑡 ≤ 𝑇

, 𝑥 = 0 and 𝑦 = 0 for 𝑡 > 𝑇

𝑇 . The incentive compatibility constraints might be rewritten as

(𝐼𝐶

) 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 𝑦 − 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 = 0,

(𝐼𝐶

) 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 𝑦 − 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 ∆𝑐 𝑞 𝑐 = 0.

Combining the two equations together we have

𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 − (1 − 𝜆 ) 𝜆 𝑦 + 𝛽 ∑

𝛿 (1 − 𝜆 ) 𝜆 −

(1 − 𝜆 ) 𝜆 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝑞 𝑐 = 0,

𝛽 ∑

𝛿 𝑓 𝑡, 𝑇 𝑦 −𝛽 ∑

𝛿 𝑓 𝑡, 𝑇 𝑦 − 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝑞 𝑐 = 0, and we have a contradiction.

Thus, the (𝐼𝐶

) is not binding. Q.E.D.

Since the (𝐼𝐶

) constraint is not binding the low type does not receive any rent and it immediately follows that 𝑥 = 0 and 𝑦 = 0 for 1 ≤ 𝑡 ≤ 𝑇, which implies that the rent of the high type in this case becomes 𝛿

𝑃

∆𝑐

𝑞 𝑐

. Replacing 𝑥 in the objective function, the principal’s optimization problem is to choose 𝑇, {𝑞 ( 𝑐 )} , 𝑞 𝑐 , {𝑦 } , {𝑞 ( 𝑐 )} and 𝑞 𝑐 to

𝑀𝑎𝑥 𝐸 𝜋 𝜛 − 𝜐𝛿 𝑃 ∆𝑐 𝑞 𝑐 subject to:

(𝐿𝐿𝑆 ) 𝑦 ≥ 0 for 𝑡 ≤ 𝑇 ,

𝐿𝐿𝐹 𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 ≥ 0.

When the (𝐼𝐶

) constraint is not binding, the claim below shows that there are no restrictions in choosing {𝑦 } except those imposed by the (𝐼𝐶

) constraint. In other words,

4

Since 𝑓 𝑡, 𝑇 < 0 for 𝑡 < 𝑇 < 𝑇

𝑇 and 𝑞 𝑐 < 𝑞 𝑐 .

5

Since 𝑓 𝑡, 𝑇 < 0 for 𝑡 < 𝑇

𝑇 , 𝑓 𝑡, 𝑇 > 0 for 𝑡 > 𝑇

𝑇 and 𝑞 𝑐 < 𝑞 𝑐 .

29

the principal can choose any combinations of nonnegative payments to the high type 𝑥 , {𝑦 } such that 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 + 𝛿 𝑃 𝑥 = 𝛿 𝑃 ∆𝑐 𝑞 𝑐 .

Labeling, {𝛼 }

, 𝛼 as the Lagrange multipliers of the constraints associated with (𝐿𝐿𝑆 ), and 𝐿𝐿𝐹

respectively, we have the following claim.

Claim D.1: If (𝐼𝐶

) is not binding, we have 𝛼 = 0 and 𝛼 = 0 for all 𝑡 ≤ 𝑇 . Proof: We can rewrite the Kuhn-Tucker conditions as follows:

ℒ

= 𝛼 − 𝛼 𝛽 𝛿 (1 − 𝜆 ) 𝜆 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ;

ℒ

= 𝑦 ≥ 0; 𝛼 ≥ 0; 𝛼 𝑦 = 0 for 1 ≤ 𝑡 ≤ 𝑇 . Suppose to the contrary that 𝛼 > 0. Then,

𝛿 𝑃 ∆𝑐 𝑞 𝑐 − 𝛽 ∑ 𝛿 (1 − 𝜆 ) 𝜆 𝑦 = 0,

and there must exist 𝑦 > 0 for some 1 ≤ 𝑠 ≤ 𝑇 . Then, we have 𝛼 = 0, which leads to a contradiction since

= 0 cannot be satisfied unless 𝛼 = 0.

Suppose to the contrary that 𝛼 > 0 for some 1 ≤ 𝑠 ≤ 𝑇 . Then, 𝛼 > 0, which leads to a

contradiction as we have just shown above. Q.E.D.

II. Optimal length of experimentation

Since 𝑇 affects the information rent of the high type, 𝑈 , there will be a distortion in the duration of the experimentation stage for both types:

ℒ

= = 0.

Since the informational rent of the high-type agent, 𝛿

𝑃

∆𝑐

𝑞 𝑐

, is non-monotonic in 𝑇, it is possible, in general, to have 𝑇 > 𝑇 or 𝑇 < 𝑇 .

III. Optimal outputs

To characterize the optimal output choices, consider now the following Kuhn-Tucker conditions for the optimization problem:

[𝑞 ( 𝑐 )] 𝜐𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 ( 𝑐 ) − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ; [𝑞 ( 𝑐 )] (1 − 𝜐)𝛽 𝛿 (1 − 𝜆 ) 𝜆 𝑉 𝑞 ( 𝑐 ) − 𝑐 = 0 for 1 ≤ 𝑡 ≤ 𝑇 ;

𝑞 𝑐

𝜐𝛿

𝑃

𝑉 𝑞 𝑐

− 𝑐

= 0;

𝑞 𝑐

(1 − 𝜐)𝛿 𝑃

𝑉 𝑞 𝑐

− 𝑐

− 𝜐𝛿

𝑃

∆𝑐

= 0.

The conditions above imply that there is no distortion in the output relative to the first-

best level after success has been observed by either type; that is 𝑉 𝑞 ( 𝑐 ) = 𝑐 for 𝜃 ∈ {𝐻, 𝐿}.

30

There is no distortion in the output relative to the first-best level after 𝑇 failures have been reported by the high type either; that is

𝑞 𝑐

= 𝑞 𝑐

.

However, 𝑉 𝑞 𝑐

= 𝑐

+

∆𝑐

, which given that function 𝑉(∙) is increasing and concave implies under production by the low type after 𝑇 failures: 𝑞 𝑐

< 𝑞 𝑐

.

Appendix F (Characterizing 𝜆)

Claim. There exists 𝝀 ∈ (𝟎, 𝟏), such that

𝒅𝝀^𝜽

> 𝟎 for 𝝀

< 𝝀 and

𝒅𝝀^𝜽

≤ 𝟎 for 𝝀

≥ 𝝀.

Proof: We prove that the monotonicity of the first-best termination date 𝑇 𝜆 depends on 𝛽 𝜆 which is increasing for small values of 𝜆 and decreasing for high values of 𝜆 with a unique point of extremum.

It is instructive to begin the proof by assuming that output after failure is exogenously fixed at some level, 𝑞 𝑐 = 𝑞 > 0. We then complete the proof for endogenous outputs.

The first-best termination date 𝑇 is then such that

𝛽 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) + 1 − 𝛽 𝜆 𝑉(𝑞 ) − 𝑐 𝑞 − 𝛾 =

𝑉(𝑞 ) − 𝑐 𝑞 or, equivalently,

𝛽 𝜆 𝑉 𝑞 (𝑐) − 𝑐𝑞 (𝑐) − 𝛽 𝜆 𝑉(𝑞 ) + 𝑐 − 𝑐 1 − 𝛽 𝜆 𝑞 − 𝛾 = 0.

First, we simplify the expression for 𝑐 1 − 𝛽 𝜆 :

𝑐 1 − 𝛽 𝜆 = 𝛽 𝑐 + 1 − 𝛽 𝑐 1 − 𝛽 𝜆 =

𝑐 + 𝛽 Δ𝑐 1 − 𝛽 𝜆 = 𝑐 1 − 𝛽 𝜆 + Δ𝑐𝛽 1 − 𝛽 𝜆 =

𝑐 1 − 𝛽 𝜆 + Δ𝑐𝛽 1 − 𝜆 ,

where the last step follows from