# OK to post homework
# RDB, 2022-02-06, HW4

# 1. I identify as etc. etc.

# 2. Done.
# 3 / 4.

# In general, profit = revenue - cost, and we say that

#   revenue = quantity * market clearing price
#      cost = quantity * marginal cost of production.

# Plugging all of this in and moving things around, the model is

#       Pi(q1, q2) = qi (a - qi - q(1 - i))

# for some (positive) constant a.

# The "equilibria" of this game depends on the exact setup.

# First, note that the function f(x) = x(c - x) has a maximum at x = c/2.

# Cournot model:

#   Both firms move "simultaneously." In this case, a point (q1, q2) is an
#   equilibrium iff q1 maximizes f(q1) = P1(q1, q2) and q2 maximizes g(q2) =
#   P2(q1, q2). These functions are, respectively,

#       f(q1) = q1(A - q1 - q2)
#       g(q2) = q2(A - q1 - q2).

#   The maximum of f occurs at q1 = (A - q2) / 2, and the maximum of g occurs
#   at q2 = (A - q1) / 2. Therefore, we require

#       q1 = (A - q2) / 2
#       q2 = (A - q1) / 2,

#   which has a unique solution of q1 = q2 = A / 3.

#   In this case, the profits of the two firms are identical:

#       P1(A/3, A/3) = P2(A/3, A/3) = A^2 / 9.

# Stackelberg model:

#   Firm one moves "first," with the knowledge that Firm 2 will pick the best
#   response to its choice. (This is really a one player game once we make that
#   assumption.)

#   Since the payoff for Firm 2 is P2(q1, q2) = q2(A - q1 - q2), the best
#   response is always q2 = (A - q1) / 2. Thus, we know that the payoff for
#   Firm 1 is

#       P1(q1, (A - q1) / 2) = q1(A - q1) / 2.

#   The best choice now is still q1 = A / 2, but the payoffs are different:

#       P1(A / 2, A / 4) = A^2 / 8
#       P2(A / 2, A / 4) = A^2 / 16.

# Monopoly model:

#   The Firms coordinate efforts to produce the monopoly quantity: A / 2. They
#   do this by setting q1 = q2 = A / 4.

#   In this case the payoffs are as follows:

#       P1(A / 4, A / 4) = P2(A / 4, A / 4) = A^2 / 8.

# 5.

# Cournot total payoff:

#       A^2 / 9 + A^2 / 9 = (2 / 9) A^2.

# Stackelberg total payoff:
#       A^2 / 8 + A^2 / 16 = (3 / 16) A^2.

# Monopoly total payoff:

#       A^2 / 8 + A^2 / 8 = (1 / 4) A^2.

# The largest total payoff comes from the Monopoly model.

# 6.

# That's enough "by-hand" mathematics for the week.

monopolyChoice := solve(diff(q * (A - q)^d, q), q);

stackelbergResponse := solve(diff(q2 * (A - q1 - q2)^d, q2), q2):
stackelbergFirstEqn := subs(q2=stackelbergResponse, q1 * (A - q1 - q2)^d):
stackelbergFirst := solve(diff(stackelbergFirstEqn, q1), q1);
stackelbergSecond := simplify(subs(q1=stackelbergFirst, stackelbergResponse));

bestResponseOne := solve(diff(q1 * (A - q1 - q2)^d, q1), q1):
bestResponseTwo := solve(diff(q1 * (A - q1 - q2)^d, q1), q1):
soln := solve({q1 = bestResponseOne, q2 = bestResponseTwo}, {q1, q2}):
nashOne := rhs(soln[1]);
nashTwo := rhs(soln[2]);