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Pareto Gains from Limiting Compensation Options

Debashis Pal
Debashis Pal
, 
David E. M. Sappington
David E. M. Sappington
 and 
Iryna Topolyan
Iryna Topolyan
   | Nov 11, 2023
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Published Online: Nov 11, 2023
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Received: Feb 06, 2023
DOI: https://doi.org/10.2478/izajole-2023-0002
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© 2023 Debashis Pal et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

Timing in the Setting with Exogenous Ability.
Timing in the Setting with Exogenous Ability.

Figure 2

Timing in the Setting with Endogenous Ability.
Timing in the Setting with Endogenous Ability.

The Effects of an SPP in Example 1.

θH θHθL \frac{{\boldsymbol {\theta _H}}}{{\boldsymbol {\theta _L}}} ϕH* {\boldsymbol {\phi _H^*}} ϕHS* {\boldsymbol {\phi _{HS}^*}} EUϕH* {\boldsymbol {EU\left( {\phi _H^*} \right)}} EUSϕHS* {\boldsymbol {E{U_S}\left( {\phi _{HS}^*} \right)}} EΠϕH* {\boldsymbol {E\Pi \left( {\phi _H^*} \right)}} EΠSϕHS* {\boldsymbol {E{\Pi _S}\left( {\phi _{HS}^*} \right)}}
0.30 1.2 0.200 0.213 0.0025 0.0026 1.12524 1.12521
0.35 1.4 0.258 0.328 0.0066 0.0078 1.12658 1.12670
0.40 1.6 0.264 0.392 0.0099 0.0131 1.1285 1.12971
0.45 1.8 0.257 0.438 0.0125 0.0180 1.13055 1.13402
0.50 2.0 0.246 0.478 0.0145 0.0227 1.13258 1.13959
0.55 2.2 0.235 0.517 0.0161 0.0272 1.13453 1.14644
0.60 2.4 0.225 0.557 0.0174 0.0316 1.13638 1.15467
0.65 2.6 0.215 0.599 0.0185 0.0361 1.13813 1.16437
0.70 2.8 0.206 0.644 0.0194 0.0406 1.13978 1.17565
0.75 3.0 0.198 0.692 0.0202 0.0453 1.14134 1.18858
0.80 3.2 0.190 0.743 0.0209 0.0502 1.14281 1.20322

Optimal Payments in the Setting of Example 1.

θH = 0.3 θH = 0.4 θH = 0.5 θH = 0.6 θH = 0.7 θH = 0.8
W_L {\boldsymbol {\underline W _L}} − 0.113 − 0.085 − 0.071 − 0.063 − 0.058 − 0.054
W¯L {\boldsymbol {\bar W_L}} 0.839 0.738 0.683 0.648 0.623 0.605
ΔL 0.952 0.823 0.754 0.711 0.681 0.659
W_H {\boldsymbol {\underline W _H}} − 0.127 − 0.149 − 0.179 − 0.212 − 0.246 − 0.280
W¯H {\boldsymbol {\bar W_H}} 0.873 0.851 0.821 0.788 0.754 0.720
ΔH 1.0 1.0 1.0 1.0 1.0 1.0
W_ {\boldsymbol {\underline W}} − 0.115 − 0.088 − 0.071 − 0.060 − 0.053 − 0.048
W¯ {\boldsymbol {\bar W}} 0.845 0.752 0.684 0.635 0.598 0.570
Δ 0.961 0.840 0.756 0.695 0.651 0.617

The Minimum θHθL \frac{{{\theta .H}}}{{{\theta .L}}} that Ensures Pareto Gains in Example 2.

k θHθL^ {\boldsymbol {\widehat {\frac{{{\theta _H}}}{{{\theta _L}}}}}} k θHθL^ {\boldsymbol {\widehat {\frac{{{\theta _H}}}{{{\theta _L}}}}}} k θHθL^ {\boldsymbol {\widehat {\frac{{{\theta _H}}}{{{\theta _L}}}}}}
0.10 1.34 0.25 1.56 1.0 2.18
0.15 1.44 0.50 1.88 2.0 2.72
0.20 1.52 0.75 2.00 5.0 3.68
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