The partial derivative of wage ($$w$$) with respect to the profit level ($$\Pi_0$$), holding employment ($$N$$) constant, is algebraically expressed as:

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

Since employment ($$N$$) is positive, this negative derivative mathematically confirms that higher profit levels correspond to lower wage levels, meaning that curves representing higher profits lie lower on the wage-employment diagram.

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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.

Algebraically Verifying Isoprofit Curve Positions Using Partial Differentiation

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

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