Dear all,
I need help in the workings of the following problem. please help! Thanks in advance.
\(\displaystyle \mbox{Maximise}\)
\(\displaystyle 0.8\, \left(\Pi_2\, -\, w_2\right)\, +\, 0.2\, \left(\Pi_1\, -\, w_1\right)\)
\(\displaystyle \mbox{subject to a 'participation constraint'}\)
\(\displaystyle 0.8\, \left(w_2^{1/2}\, -\, 1\right)\, +\, 0.2\,\left(w_1^{1/2}\, -\, 1\right)\, \geq\, 2\)
\(\displaystyle \mbox{and an 'incentive-compatibility constraint'}\)
\(\displaystyle 0.8\, \left(w_2^{1/2}\, -\, 1\right)\, +\, 0.2\, \left(w_1^{1/2}\, -\, 1\right)\, \geq\, 0.3\, w_2^{1/2}\, +\, 0.7\, w_1^{1/2}\)
\(\displaystyle \mbox{This maximisation problem is easy to solve because both constraints}\)
\(\displaystyle \mbox{are satisfied with equality. We get a system of two equations in }\, w_1\)
\(\displaystyle \mbox{and }\, w_2.\, \mbox{ Solving, we obtain }\, w_1\, =\, 1.96\, \mbox{ and }\, w_2\, =\, 11.56.\)
I need help in the workings of the following problem. please help! Thanks in advance.
\(\displaystyle \mbox{Maximise}\)
\(\displaystyle 0.8\, \left(\Pi_2\, -\, w_2\right)\, +\, 0.2\, \left(\Pi_1\, -\, w_1\right)\)
\(\displaystyle \mbox{subject to a 'participation constraint'}\)
\(\displaystyle 0.8\, \left(w_2^{1/2}\, -\, 1\right)\, +\, 0.2\,\left(w_1^{1/2}\, -\, 1\right)\, \geq\, 2\)
\(\displaystyle \mbox{and an 'incentive-compatibility constraint'}\)
\(\displaystyle 0.8\, \left(w_2^{1/2}\, -\, 1\right)\, +\, 0.2\, \left(w_1^{1/2}\, -\, 1\right)\, \geq\, 0.3\, w_2^{1/2}\, +\, 0.7\, w_1^{1/2}\)
\(\displaystyle \mbox{This maximisation problem is easy to solve because both constraints}\)
\(\displaystyle \mbox{are satisfied with equality. We get a system of two equations in }\, w_1\)
\(\displaystyle \mbox{and }\, w_2.\, \mbox{ Solving, we obtain }\, w_1\, =\, 1.96\, \mbox{ and }\, w_2\, =\, 11.56.\)
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