Answer
a) As $N$ increases, the demand curve for each firm shrinks (or decreases).
b) $25/N$ is the quantity.
c) $75/N$ is the price.
d) $\frac{1250}{N^2} - 50$ is the profit.
e) 5 firms
Work Step by Step
a) If we let $N=3$, then we have
$D = 100/3 -P$ (for each of the three firms)
Since each firm has the same demand curve, and if we add the demand curves together, we have a total demand curve of $100-3P=D$. Since we have a finite demand curve, as $N$ increases, the individual demand curves for each firm shrinks.
b)
$MR = MC$
$100/N -2Q = 2Q$
$100/N = 4Q$
$100/N*1/4 = 4Q*1/4$
$100/4N = Q$
$25/N = Q$
c)
$25/N = 100/N - P$
$25/N+P-25/N = 100/N - P+P-25/N$
$P = 75/N$
d)
profit = revenue - cost
total cost = $50 + Q^2$
$TC = 50 + (25/N)^2$
$TC = 50 + \frac{625}{N^2}$
revenue = price * quantity
$R = 75/N * 25/N$
$R = \frac{1875}{N^2}$
profit = revenue - cost
$R = \frac{1875}{N^2} - (50 + \frac{625}{N^2})$
$R = \frac{1875}{N^2} - 50 - \frac{625}{N^2}$
$R = \frac{1250}{N^2} - 50$
e)
Since the market will exist with firms making close to no profit, we set profit to zero (and solve for $N$).
$R = \frac{1250}{N^2} - 50$
$0 = \frac{1250}{N^2} - 50$
$50 = \frac{1250}{N^2}$
$50N^2 = 1250$
$50N^2/50 = 1250/50$
$N^2 = 25$
$\sqrt {N^2} = \sqrt {25}$
$N = 5$
