(Solved): Could you please give more explanation of part B, I asked once, but I still cannot understand the so ...

2. An economy has two kinds of consumers and two goods. Type $A$ consumers have utility functions $U\left(x_1,x_2\right)=4x_1-\left(x_1^2/2\right)+x_2$ and Type $B$ consumers have utility functions $U\left(x_1,x_2\right)=2x_1-\left(x_1^2/2\right)+x_2$. Consumers can only consume nonnegative quantities. The price of good 2 is $\$1$ and all consumers have income of $\$100$. There are $N$ type $\mathrm{A}$ consumers and $N$ type $B$ consumers. (a) Suppose that a monopolist can produce good 1 at a constant unit cost of $c$ per unit and cannot engage in any kind of price discrimination. Find its optimal choice of price and quantity. For what value of $c$ will it choose to sell to both types of consumers? (b) Suppose that the monopolist uses a "two-part tariff" where a consumer must pay a lump sum $k$ in order to be able to buy anything at all. A person who has paid the lump sum $k$ can buy as much as he likes at price of $p$ per unit purchased. Consumers are not able to resell good 1 . For $p<4$, what is the highest amount of $k$ that a type $A$ is willing to pay for the privilege of buying at price $p$ ? If a type A does pay the lump sum $k$ to buy at price $p$, how many units will he demand? Describe the demand function for good 1 by each type of consumers as a function of $p$ and $k$. (c) If the monopolist can separate the two types of consumers and engage in price discrimination, what would be the profit-maximizing choices of $p$ and $k$ for each type of consumers?