Second-Derivative Test Showing Profit-Maximizing Output is Inefficiently High
The second derivative of the social utility function can be used to demonstrate that a plantation's profit-maximizing output () is excessively high when negative externalities exist. At the output level , where the first derivative of the utility function is negative, the second derivative is also negative, under the condition that the marginal external cost (MEC) increases with output. Since both the function's slope and its rate of change are negative, it mathematically confirms that output must be reduced from to satisfy the first-order condition for a Pareto-efficient allocation.
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Second-Derivative Test Showing Profit-Maximizing Output is Inefficiently High
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