Suppose that Bill, George, and $\mathrm { Al }$ constitute the entire market for consumers of national defense. Each man has an identical demand curve for national defense, which can be expressed as $P =$ $50 - Q$. Suppose that the marginal cost for national defense can be expressed as $M C = \$ 30$. What is the optimal quantity of national defense?
A) 150 units.
B) 60 units
C) 40 units.
D) 20 units.

Question

Quizplus · Accepted Answer

To find the optimal quantity of national defense, we need to use the formula for marginal cost (MC) and the demand curve for national defense.

First, we need to find the quantity at which marginal cost equals the demand price:

\begin{aligned} MC &= P\ \$30 &= 50 - Q\ Q &= 20. \end{aligned}

So the quantity demanded at the marginal cost equals the demand price is 20 units.

Next, we need to find the total quantity demanded by Bill, George, and Al at this price:

\begin{aligned} P &= 50 - Q\ P &= 50 - 20\ P &= \$30. \end{aligned}

So the price is $30 per unit. Each of the three consumers demand:

\begin{aligned} Q_B &= Q_G &= Q_A = 50 - 30\ Q_B &= Q_G &= Q_A = 20. \end{aligned}

Total quantity demanded is $20 	imes 3 = 60$ units. However, this is not the optimal quantity because the marginal cost is less than the price at this quantity.

We need to keep increasing the quantity until the marginal cost equals the price. So we increase the quantity by one unit and see if MC equals P.

\begin{aligned} MC &= P\ \$30 &= 50 - Q\ Q &= 20. \end{aligned}

This is still less than the quantity demanded at $P = \$30$, so we need to keep increasing the quantity.

\begin{aligned} MC &= P\ \$30 &= 50 - Q\ Q &= 20. \end{aligned}

Now the quantity demanded at the price $P = \$30$ equals the quantity at which the marginal cost equals the price. This is the optimal quantity of national defense.

Therefore, the optimal quantity of national defense is 40 units.

Suppose That Bill, George, And Al\mathrm { Al }Al Constitute the Entire Market for Consumers of National Defense

Suppose That Bill, George, And $\mathrm { Al }$ Constitute the Entire Market for Consumers of National Defense