aggregate of a continuum of individual goods, ct = [R0 ct(z)(ε-1Pεdz]P(ε-1). Households
distribute their expenditure efficiently across these goods, resulting in constant-elasticity
demands for individual products from each of the two types of firms which they will
encounter in the equilibrium below:
cj,t = P~j^ ct, j = 0,1∙ (3)
The subscript j in (3) denotes the age of the nominal price, so that P0,t is the price set by
firms in period t and P1,t is the price set by firms one period previously (P1,t = P0,t-1).
Likewise, cj,t is the period-t demand for goods produced by a firm that set its price in
period t - j . The price level which enters in these demands takes the form
1 1 1 1 ι
Pt = [2 Pl- ε + 2 Pι1,-ε]1-ε. (4)
We assume that households also hold money to finance expenditure, according to
Mt
0 Pt(z)ct (z) dz
(5)
so that our model imposes a constant, unit velocity, in common with many macroeconomic
analyses.1 We adopt this specification because it allows us to abstract from all the wealth
and substitution effects that normally arise in optimizing models of money demand, so
as to focus on the consequences of price-stickiness. With constant-elasticity demands for
each good, the money-demand specification in (5) implies
Mt = Pt ct .
(6)
Since this is a representative agent model and since no real accumulation is possible
given the technologies described below, we are not too explicit about the consumption-
saving aspect of the household’s problem; it will be largely irrelevant in general equilibrium
except for asset-pricing. We simply assume that there is a Lagrange multiplier that takes
the form
∂u(ct,lt) 1
λt = --â---- = -, (7)
∂ct ct
and that households equate the marginal rate of substitution between leisure and con-
sumption to the real wage rate prevailing in the competitive labor market, i.e.,
∂u(ct,lt)∕∂ lt
wt = ∂ u(ct,lt)∕∂ ct = χct.
(8)
1 We think of this quantity equation as the limiting version of a standard money demand function which
occurs as the own return on money is raised toward the nominal interest rate (see King and Wolman [1999]
for some additional discussion).