Question

Consider the following consumption decision problem. A consumer lives for two periods and receives income of y in each period. She chooses to consume c1 units of a good in period 1 and c2 units of the good in period 2. The price of the good is one. The consumer can borrow or invest at rate r. The consumer’s utility function is: U = ln(c1) + δ ln(c2), where δ > 0.

a. Derive the optimal consumption in each period?

b. How does optimal consumption in each period vary with the interest rate? Explain why.

c. Use your answer in part (a) to derive a condition that determines whether the consumer saves in the first period? Interpret this condition.

Answer #1

Consider an individual who lives two periods. He works in both
periods and receives a labor income of 200 euros in the first
period and 220 euros in the second. The interest rate of the
economy is 10%. The consumption in period 1 is c1, and in period 2
it is c2. The price of the consumption good is 1 in both periods.
The utility function of this individual is
U(c1,c2
)=c11/2c21/2. Suppose
there is a proportional tax on labor...

Consider an individual who lives two periods. He works in both
periods and receives a labor income of 200 euros in the first
period and 220 euros in the second. The interest rate of the
economy is 10%. The consumption in period 1 is c1, and in period 2
it is c2. The price of the consumption good is 1 in both periods.
The utility function of this individual is
U(c1,c2
)=c11/2c21/2. Suppose
there is a proportional tax on labor...

Imagine an individual who lives for two periods. The individual
has a given pattern of endowment income (y1 and
y2) and faces the positive real interest rate, r.
Lifetime utility is given by U(c1, c2)=
ln(c1)+β ln(c2)
Suppose that the individual faces a proportional consumption tax
at the rate Ԏc in each period. (If the individual
consumes X in period i then he must pay XԎc to the
government in taxes period). Derive the individual's budget
constraint and the F.O.C...

Assume the representative consumer lives in two periods and his
preferences can be described by the utility function U(c,c′)=c1/3
+β(c′)1/3, where c is the current consumption, c′ is next period
consumption, and β = 0.95. Let’s assume that the consumer can
borrow or lend at the interest rate r = 10%. The consumer receives
an income y = 100 in the current period and y′ = 110 in the next
period. The government wants to spend G = 30 in...

A consumer’s consumption-utility function for a two-period
horizon (t = 1, 2) is given by U(c1,c2) = ln(c1)+ln(c2). The
consumer’s income stream is y1 = $1500 and y2 = $1080, and the
market rate of interest is 8%. Calculate the optimal values for c1
and c2 that maximize the consumer’s utility

Beta lives for two periods. In period 1, Beta works and earns a
total income of $2, 000. If she consumes $c1 in period 1, then she
deposits her savings of 2, 000 − c1 dollars in a bank account that
gives her an interest rate of 10% per period. (Notice that Beta is
not able to borrow in period 1, so c1 ≤ 2, 000.) In period 2, Beta
leads a retired life and receives $110 in social-security income....

4.5 Susan is certain to live just two periods and
receives an income of 10,000 in the first period, and 15,000 in the
second. She has no other assets. The real interest rate is
8%.
(a) As she begins the first period, what is the present
value of her lifetime resources?
(b) IF she choose to consume the exact same amount in
both periods (c1 = c2), what would be her consumption in the first
(and second) period? SHOW YOUR...

Beta lives for two periods. In period 1, Beta works and earns a
total income of $2, 000. If she consumes $c1 in period 1, then she
deposits her savings of 2, 000 − c1 dollars in a bank account that
gives her an interest rate of 10% per period. (Notice that Beta is
not able to borrow in period 1, so c1 ≤ 2, 000.) In period 2, Beta
leads a retired life and receives $110 in social-security income....

A consumer likes two goods; good 1 and good 2. the consumer’s
preferences are described the by the cobb-douglass utility
function
U = (c1,c2) =
c1α,c21-α
Where c1 denotes consumption of good 1, c2
denotes consumption of good 2, and parameter α lies between zero
and one; 1>α>0. Let I denote consumer’s income, let
p1 denotes the price of good 1, and p2
denotes the price of good 2. Then the consumer can be viewed as
choosing c1 and c2...

A consumer is endowed with income M1 in the present and M2 in
the future, with M1 = M2 > 0. C1 and C2 are consumption in the
present and future, respectively. The consumer has convex
indifference curves, described by U(C1, C2), and can borrow and
save at the nominal rate of interest of r > 0. (i) Use an
isoquant diagram to carefully illustrate the optimal consumption
choice when the consumer is a lender in the present (6 points)....

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