4 BauTobOneAsset

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A Baumol-Tobin Money Demand Model.
Juanpa Nicolini
October 30, 2017
The Model
• Households have an endowment of 1 unit of time.
• Choose the number of times they make portfolio adjustments during
a period.
• If the household makes nt exchanges between bonds for transactional
assets within the period, it must pay a cost of c(nt), in units of time.
• The Baumol-Tobin case obtains when σ = 1, so the cost is a linear
function of the number of trips to the bank.
• The rest of time is devote to labor.
• Production will therefore be
zt(1− c(nt)) = xt
• Continuum of identical households with the common preferences
∞∑
t=0
βtU(xt).
• Goods are not storable.
• The restrictions
Mt +Bt ≤Wt
Wt+1 ≤Mt(1− i
m) +Bt(1 + it) + zt(1− c(nt))Pt − Ptxt + Tt
where im is the cost of using money, and
Ptxt − ntMt ≤ 0
• first order conditions
ct : β
tU ′(xt) = λtPt + δtPt
nt : δtMt = λtztc
′(nt)Pt
Mt : λt(1− i
m) + δtnt = ωt
Bt : ωt = λt(1 + it)
Wt+1 : λt = ωt+1
• Use Eq 4 in Eq 3 and combine with Eq2.
δtMt = λtztc
′(nt)Pt
λt(1− i
m) + δtnt = λt(1 + it)
• Use first to eliminate δt in second
λt(1− i
m) + nt
λtztc
′(nt)Pt
Mt
= λt(1 + it)
or
(1− im) + nt
ztc
′(nt)Pt
Mt
= (1 + it)
or
nt
ztc
′(nt)Pt
Mt
= it − i
m
• Recall that
Pt
Mt
=
nt
xt
so
ztc
′(nt)
n2t
xt
= it − i
m
or
c′(nt)
n2t
(1− c(nt)
= it − i
m