Common Notation for General Equilibrium with Asset Markets

Environment

T = set of time periods (usually, t = 0 is present; t = 1, 2, .. is future, t = T is doomsday).
S = set of states (#S = S)
H = set of households (#H = H)
F = set of financial assets (#F = F)
K = set of firms (#K = K)
n = number of physically differentiated goods.
p_is = spot price of good i in state s.
p_s = spot prices in state s
p = [p₀, p₁, .., p_S] = vector of spot prices.

Households

B_A = Arrow-Debreu budget set.
x_is^h = amount of commodity i received by household h Î H in state s Î S.
x_s^h = vector of commodities assigned to household h Î H in state s Î S.
x^h = [x₀^h, x₁^h, x₂^h, .., x_S^h] = consumption plan of household h Î H.
u^h: R^n(S+1) ® R - utility function of household h.
y ^h: R^n(s+1)´ R ® R - indirect uility function of household h.
p _s^h: probability of state s as believed by household h.
e^h = [e₀^h, e₁^h, e₂^h, .., e_S^h] = endowment pattern of household h Î H.
l _s^h = Lagrangian multiplier of sth constraint of household h's optimization problem

B_R = Radner budget set.
m _s^h = marginal valuation of state s by household h, m _s^h = p _s^h(¶ u_s^h/¶ x_s^h) = (p _s^h/l ₀^h)(¶ y _s^h/¶ m_s^h)
m ^h = [1, m ₁^h, m ₂^h, .., m _S^h] = vector of state prices.
m ^h_-0 = [m ₁^h, m ₂^h, .., m _S^h] = vector of state prices minus the first argument.
t _s^h = net transfer of purchasing power to/from state s Î S+1 by household h.
W = hyperplane denoting asset market constraint.
W^{^} = hyperplane perpindicular to W.

Assets

a_f^h = amount of asset f Î F purchased by household h.
a^h = [a₁^h, a₂^h, .., a_F^h] = portfolio purchased by household h
q_f = price of asset f Î F.
r _fs = payoff of asset f in state s.
r_fs = monetary return of asset f in state s, r_fs = p_{sr fs}
R_fs = financial return of asset f in state s, R_fs = r_fs/q_f
r = risk-free return on a bond.
r_f: S ® R = [r_f1, r_f2, .., r_fS] = random variable denoting returns of asset f Î F (also denoted V_f)
r_s: F ® R = [r_1s, r_2s, .., r_Fs] = distribution of returns across assets in state s Î S (also denoted V_s)
V = S ´ F matrix of returns with r_f, f = 1, .., F in columns and r_s, s = 1, .., S as rows.

Firms

Y^k = production set of firm k Î K.
y^k = [y₀^k, y₁^k, .., y_S^k] = production plan of firm k.
T^k(y^k) = transformation function of firm k.
v^k = market value of firm k
d _s^k = dividend paid by firm k in state s.
b^k = portfolio purchased by firm k (= negative of debt issued by firm k).
q ^hk = percentage of firm k owned by household h.