The cost of controlling emissions at a firm is given by...

The cost of controlling emissions at a firm is given by $$C(q)=4,000+100 q^{2}$$ where $q$ is the reduction in emissions (in pounds of pollutant per day) and $C$ is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $$\$ 500$$ per pound of pollutant removed. How many pounds of pollutant should the firm remove each day in order to minimize $n e t$ cost (cost minus subsidy)?

The cost of controlling emissions at a firm is given by
$$C(q)=4,000+100 q^{2}$$
where $q$ is the reduction in emissions (in pounds of pollutant per day) and $C$ is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $$\$ 500$$ per pound of pollutant removed. How many pounds of pollutant should the firm remove each day in order to minimize $n e t$ cost (cost minus subsidy)?

Key Concepts

Optimization and Minimization

Optimization in economics involves finding the level of an activity that either minimizes costs or maximizes benefits. This process often involves setting up an objective function—in this case, the net cost after accounting for subsidies—and using mathematical techniques like differentiation to find the minimum or maximum point.

Marginal Analysis

Marginal analysis involves comparing the additional (marginal) cost of an increase in an activity with the additional benefit. When the marginal cost equals the marginal benefit (or adjusted for subsidies), the firm can achieve an economically optimal decision, which is central to making cost-effective operational decisions.

Cost Functions

A cost function represents the total cost incurred by a firm in relation to the level of a particular activity, such as reducing emissions. It is used to model and analyze how costs change as the level of output or control effort changes, and is fundamental in determining efficient levels of production or control measures.

Subsidies in Economic Analysis

Subsidies are financial incentives provided by a government to encourage certain behaviors such as pollution reduction. They effectively lower the net cost to the firm for undertaking these actions, and their impact must be incorporated into economic decision-making to determine the optimal level of activity.

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