Suppose that the total benefit and total cost from an activity are, respectively, given by the following equations: B(Q) = 150 + 28Q – 5Q2 and C(Q) = 100 + 8Q. (Note: MB(Q) = 28 – 10Q and MC(Q) = 8.)
Write out the equation for the net benefits.
What are the net benefits when Q = 1? Q = 5?
Write out the equation for the marginal net benefits.
What are the marginal net benefits when Q = 1? Q = 5?
What level of Q maximizes net benefits?
At the value of Q that maximizes net benefits, what is the value of marginal net benefits?
The Correct Answer and Explanation is:
Net Benefits Equation
The net benefit from an activity is simply the difference between the total benefit (B(Q)) and the total cost (C(Q)):
$$
\text{Net Benefit (NB)}(Q) = B(Q) – C(Q)
$$
Given the equations for total benefit and total cost:
- $B(Q) = 150 + 28Q – 5Q^2$
- $C(Q) = 100 + 8Q$
Substitute these into the net benefits equation:
$$
\text{NB}(Q) = (150 + 28Q – 5Q^2) – (100 + 8Q)
$$
Simplify:
$$
\text{NB}(Q) = 150 + 28Q – 5Q^2 – 100 – 8Q
$$
$$
\text{NB}(Q) = 50 + 20Q – 5Q^2
$$
Thus, the net benefits equation is:
$$
\text{NB}(Q) = 50 + 20Q – 5Q^2
$$
Net Benefits at $Q = 1$ and $Q = 5$
Now, substitute $Q = 1$ and $Q = 5$ into the net benefits equation.
For $Q = 1$:
$$
\text{NB}(1) = 50 + 20(1) – 5(1)^2 = 50 + 20 – 5 = 65
$$
For $Q = 5$:
$$
\text{NB}(5) = 50 + 20(5) – 5(5)^2 = 50 + 100 – 125 = 25
$$
Thus, the net benefits are:
- When $Q = 1$, the net benefit is 65.
- When $Q = 5$, the net benefit is 25.
Marginal Net Benefits Equation
The marginal net benefit is the derivative of the net benefit function with respect to $Q$:
$$
\text{Marginal Net Benefit (MNB)}(Q) = \frac{d}{dQ}\left(50 + 20Q – 5Q^2\right)
$$
Differentiate each term:
$$
\text{MNB}(Q) = 20 – 10Q
$$
Thus, the marginal net benefits equation is:
$$
\text{MNB}(Q) = 20 – 10Q
$$
Marginal Net Benefits at $Q = 1$ and $Q = 5$
Substitute $Q = 1$ and $Q = 5$ into the marginal net benefits equation.
For $Q = 1$:
$$
\text{MNB}(1) = 20 – 10(1) = 20 – 10 = 10
$$
For $Q = 5$:
$$
\text{MNB}(5) = 20 – 10(5) = 20 – 50 = -30
$$
Thus, the marginal net benefits are:
- When $Q = 1$, the marginal net benefit is 10.
- When $Q = 5$, the marginal net benefit is -30.
Maximizing Net Benefits
To maximize net benefits, set the marginal net benefit equal to zero:
$$
20 – 10Q = 0
$$
Solve for $Q$:
$$
10Q = 20 \quad \Rightarrow \quad Q = 2
$$
Thus, the level of $Q$ that maximizes net benefits is $Q = 2$.
Marginal Net Benefits at $Q = 2$
Substitute $Q = 2$ into the marginal net benefits equation:
$$
\text{MNB}(2) = 20 – 10(2) = 20 – 20 = 0
$$
Thus, the value of marginal net benefits at the value of $Q$ that maximizes net benefits is 0.
Explanation
The net benefits equation is derived from the difference between total benefits and total costs, which shows the economic gain from the activity at any given level of $Q$. By finding where the marginal net benefit equals zero, we determine the point where the net benefits are maximized, as the marginal net benefit represents the change in net benefits for each additional unit of $Q$. The optimal level of $Q$ maximizes net benefits and ensures that no additional benefit is gained from increasing $Q$ further, making it the most efficient output level.