Monetary Policy, Inflation, and the Business Cycle Chapter 2 A ...

c) Discuss how utility depends on the two parameters describing monetary
policy, m and 2 m (recall that the nominal interest rate is constrained to be
non-negative, i.e., Q t 1 for all t). Show that the optimal policy must satisfy
the Friedman rule (i t = 0 all t) and discuss alternative ways of supporting
that rule in equilibrium.
5. A Shopping Time Model (based on Walsh (2003)).
Assume that the transactions technology is such that consuming C t requires
a quantity of shopping time Nt s = s C t ; Mt
P t
, where s c > 0 and s m 0.
Hence the amount of time diverted from leisure is given by N t +Nt s , where N t
denotes hours of work. Let the original period utility be given by V (C t ; L t )
where L t = 1 N t Nt s denotes leisure.
a) Derive the condition determining the optimal allocation of time.
b) Derive the implied utility function in terms of consumption, hours and
real balances, and discuss its properties.
6. A Model with Cash and Credit Goods
Assume that the utility of the representative household is given by:
V (C 1t ; C 2t ; N t ) (47)
where C 1t denotes consumption of a “cash-good”(i.e., a good that requires
cash in order to be purchased), C 2t is consumption of a “credit-good”( which
does not require cash), and N t is labor supply. For simplicity, let us assume
that the price of the two goods is identical and equal to P t (this will be
the case if the production function of the representative …rm is given by
Y 1t + Y 2t = N t and there is perfect competition). Purchases of cash-goods
have to be settled in cash, whereas credit goods can be …nanced by issuing
one-period riskless nominal bonds.
The budget constraint is given by
P t (C 1t + C 2t ) + Q t B t + M t = B t 1 + M t 1 + W t N t + T t
Finally, the CIA constraint is given by
P t C 1t M t
1 + T t
where, in equilibrium, T t = M t , i.e. transfers to households correspond
to money tranfers made by the central bank, and which consumers take as
given. For simplicity we assume no uncertainty.
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