Problem Set 5
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, October 15
This problem set asks you to solve a series of dynamic problems that characterize the optimal
lending and borrowing behavior of individual consumers, the determination of the equilib
rium interest rate, and the link between equilibrium and optimal resource allocations in an
economy in which, for simplicity, it is assumed that all agents live for only two periods. Each
consumer is “young” during period
t
= 0 and “old” during period
t
= 1
.
Consumers are of
two types, called “lenders” and “borrowers” for reasons that will (hopefully) become clear
when their individual optimization problems are described next.
1. Optimal Lending
Consider first the behavior of a “lender” who receives an endowment consisting of one unit
of the economy’s single consumption good during period
t
= 0 when he or she is young.
Let
c
L
0
denote the consumption of this agent during period
t
= 0 when young and let
s
L
denote the saving of this agent during period
t
= 0 when young. Assume that this lender
earns interest on his or her savings at the rate
r
between periods
t
= 0 and
t
= 1.
This
lender receives no endowment during period
t
= 1, and hence must finance consumption
c
L
1
when old exclusively from his or her savings. Assume that this consumer has an additively
timeseparable utility function with singleperiod utility that is logarithmic in form and that
utility at when old is discounted relative to utility when young using the discount factor
β
,
which satisfies 0
< β <
1. Now the lender’s dynamic optimization problem can be stated
formally as:
max
c
L
0
,c
L
1
,s
L
ln(
c
L
0
) +
β
ln(
c
L
1
) subject to 1
≥
c
L
0
+
s
L
and (1 +
r
)
s
L
≥
c
L
1
.
Note that nonnegativity conditions are not imposed on any of the choice variables, since the
form of the utility function implies that the consumer will always want to consume at least
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 Fall '09
 IRELAND
 Economics, Debt, Interest, Mortgage loan, representative, The Borrowers

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