1Cademy - Algebraically Verifying Isoprofit Curve Positions Using Partial Differentiation

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Algebraically Verifying Isoprofit Curve Positions Using Partial Differentiation

The algebraic verification that lower isoprofit curves correspond to higher profit levels is achieved by taking the partial derivative of the isoprofit equation with respect to the profit level ( $\Pi_0$ ). Rearranging the isoprofit equation to express wage ( $w$ ) as a function of employment ( $N$ ) gives:

$w = R - \frac{\Pi_0}{N}$

where $R$ is the revenue per worker. Partially differentiating $w$ with respect to $\Pi_0$ while holding $N$ constant yields:

$\frac{\partial w}{\partial \Pi_0} = -\frac{1}{N}$

Since employment ( $N$ ) is strictly positive ( $N > 0$ ), the derivative is negative ( $-\frac{1}{N} < 0$ ). This confirms that a higher profit level corresponds to a lower wage for any given level of employment.