Question

Imagine an individual who lives for two periods. The individual
has a given pattern of endowment income (y_{1} and
y_{2}) and faces the positive real interest rate, r.

Lifetime utility is given by U(c_{1}, c_{2})=
ln(c_{1})+β ln(c_{2})

Suppose that the individual faces a proportional consumption tax
at the rate Ԏ_{c} in each period. (If the individual
consumes X in period i then he must pay XԎ_{c} to the
government in taxes period). Derive the individual's budget
constraint and the F.O.C for the optimal consumption profile. What
are the optimal values of consumption in the two periods? What is
savings?

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...

i just uploaded another screenshot as a correction for below
question
【 4 】 Consider an individual who lives
for two periods. The individual has no initial wealth and earns
(exogenous) labor incomes of amounts Y1 and Y2 in
the two periods. The individual can borrow and lend at a fixed
interest rate r. The individual’s lifetime utility
function is given by U = ln C1 +
1 ln C2, where ρ is the rate of time
preference.
Also consider...

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?...

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....

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....

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...

Tom has preferences over consumption and leisure of the
following form: U = ln(c1)+ 2 ln(l)+βln(c2), where ct denotes the
stream of consumption in period t and l, hours of leisure. He can
choose to work only when he is young. If he works an hour, he can
earn 10 dollars (he can work up to 100 hours). He can also use
savings to smooth consumption over time, and if he saves, he will
earn an interest rate of 10%...

Question 2: Consumption Decisions
Suppose a person's life is divided into two main blocks, periods 1
and 2. The consumer does not desire
to perfectly smooth consumption over the two periods. In
particular, preferences are such that c2 = 0:5 c1.
Income in the two periods is equal to y1 = 500 and y2 = 1000, and
income taxes are proportional 1 = 50%
and 2 = 50%. The real interest rate is r = 0%.
(a) What is the...

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