Imagine a world with two goods: good x and a currency m. There are 1,000 identical consumers in this economy, each with an income of y and all having the following preferences: Ui(xi, mi) = 1, 000x_I 10x_i² + m_i. Suppose that the price of good x is p and that the price of cash m is 1.

a) What is the Walrasian demand, xi(p, y), of a consumer i? Assume an interior solution (ie that given by the first-order conditions).

b) Explain in words in which cases we would not have an internal solution?

c) What would be the quantity of good x consumed in such a case?

d) What is the market demand, X^d (p) for good x?

Now suppose that good x ext produced by an industrial sector with firms all having identical technology. The cost function of each of these firms is: c(q_j ) = q²_j + 400

Let us first assume that the sector is made up of firms in pure and perfect competition.

e) What is the long-term equilibrium price in the market?

f) What is the total quantity produced and consumed?

g) How many firms are there in this industrial sector?

Now suppose that the market is made up of only one firm which behaves as a monopolist.

h) What is the price charged by this firm?

i) What is the total quantity produced and consumed?

Let us now assume that the market is only made up of n firms which compete in Cournot style.

j) What is the price charged by these firms?

k) What is the total quantity produced and consumed?

l) If we assume free entry and exit from this oligopolistic industrial sector, what will be the number of firms, the price and the total production in the long-term equilibrium?

please give details and to not use A.I. thanks