Common Notation for General Equilibrium with Asset Markets
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.
pis = spot price of good i in state s.
ps = spot prices in state s
p = [p0, p1, .., pS] = vector of spot prices.
BA = Arrow-Debreu budget set.
xish = amount of commodity i received by household h ﾎ H in state s ﾎ S.
xsh = vector of commodities assigned to household h ﾎ H in state s ﾎ S.
xh = [x0h, x1h, x2h, .., xSh] = consumption plan of household h ﾎ H.
uh: Rn(S+1) ｮ R - utility function of household h.
y h: Rn(s+1)ｴ R ｮ R - indirect uility function of household h.
p sh: probability of state s as believed by household h.
eh = [e0h, e1h, e2h, .., eSh] = endowment pattern of household h ﾎ H.
l sh = Lagrangian multiplier of sth constraint of household h's optimization problem
BR = Radner budget set.
m sh = marginal valuation of state s by household h, m sh = p sh(ｶ ush/ｶ xsh) = (p sh/l 0h)(ｶ y sh/ｶ msh)
m h = [1, m 1h, m 2h, .., m Sh] = vector of state prices.
m h-0 = [m 1h, m 2h, .., m Sh] = vector of state prices minus the first argument.
t sh = net transfer of purchasing power to/from state s ﾎ S+1 by household h.
W = hyperplane denoting asset market constraint.
W^ = hyperplane perpindicular to W.
afh = amount of asset f ﾎ
F purchased by household h.
ah = [a1h, a2h, .., aFh] = portfolio purchased by household h
qf = price of asset f ﾎ F.
r fs = payoff of asset f in state s.
rfs = monetary return of asset f in state s, rfs = psr fs
Rfs = financial return of asset f in state s, Rfs = rfs/qf
r = risk-free return on a bond.
rf: S ｮ R = [rf1, rf2, .., rfS] = random variable denoting returns of asset f ﾎ F (also denoted Vf)
rs: F ｮ R = [r1s, r2s, .., rFs] = distribution of returns across assets in state s ﾎ S (also denoted Vs)
V = S ｴ F matrix of returns with rf, f = 1, .., F in columns and rs, s = 1, .., S as rows.
Yk = production set of firm k ﾎ K.
yk = [y0k, y1k, .., ySk] = production plan of firm k.
Tk(yk) = transformation function of firm k.
vk = market value of firm k
d sk = dividend paid by firm k in state s.
bk = portfolio purchased by firm k (= negative of debt issued by firm k).
q hk = percentage of firm k owned by household h.