Cobb-Douglas Effort Model

Interactive companion to the simplified identical-employees case

$$e^* = \left(\frac{b\,p\,\gamma\, N^{\gamma}}{k}\right)^{\!\frac{1}{2-\gamma}}, \qquad \gamma \in (0,1]$$

Adjust parameters

Bonus $b$ 1.5

Probability $p$ 0.50

Effort cost $k$ 3.0

Team size $N$ 10

Elasticity $\gamma$ 0.60

Optimal effort $e^*$

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$\mathbb{P}(R=1)$

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$= p \cdot (N e^*)^{\gamma}$

Marginal contribution

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$\partial\mathbb{P}/\partial e_i$

Utility curve $U(e)$ — maximum at $e^*$

Red marker = optimal effort $e^*$. Dashed vertical line shows the maximum.

$e^*$ as a function of team size $N$

Marginal contribution $\partial\mathbb{P}/\partial e_i$ vs $N$

$e^*$ as a function of bonus $b$

$e^*$ as a function of effort cost $k$

$e^*$ rises with $N$ and $b$, falls with $k$. The marginal contribution curve $\partial\mathbb{P}(R=1)/\partial e_i = p\,\gamma\,N^{\gamma-1} e^{*\gamma}/N$ always decreases with $N$ — formalising team moral hazard.

Sensitivity of $e^*$ to ±50% parameter change (tornado chart)

Key comparative statics

Parameter	Effect on $e^*$	Interpretation