Econ 482 Advanced Macroeconomic Theory II Assignment

1. Consider a three-period model we studied in class and assume a quadratic-utility con- Econ 482 Advanced Macroeconomic Theory II Assignment
sumer whose income is certain at times 0 and 2 but uncertain at time 1: y0, ̃y1, y2.

At time 1, income is y1(l) with probability 1
2
and y1(h) with probability 1
2

. Con-
sumer is allowed to borrow at time 0 but borrowing may not be allowed at time 1.

y0 = E[ ̃y1] = y2 = 2, y1(h) = 4, y1(l) = 0, r = 0, β = 1. Solve the problem.
(a) Assume that borrowing is allowed both at time 0 and time 1. Set up and solve a
consumer’s optimization problem.
(b) Assume that borrowing is not allowed at time 1, s1(h) ≥ 0 and s1(l) ≥ 0, but is
allowed at time 0. Set up and solve a consumer’s optimization problem.

2. Consider the following example we studied in class. Consumer has quadratic prefer-
ences and cares about consumption over two periods:

U(c0, c1) = −
1
2
(c0 − c ̄)
2 − β
1
2
(c1 − c ̄)
2
.

Assume that the real interest rate, r, is zero, and the time discount factor, β equals 1.
(a) Consumer’s disposable income in period 0 equals 3, and in period 1 equals 3.
There’s no uncertainty. Find the optimal consumption levels in periods 0 and 1,
and the optimal savings.

(b) Assume now that period 0 income stays at 3, while period 1 income is uncer-
tain. There’re two possible states of nature that might realize in period 1—with

probability π = 0.5, income will equal 0 in period 1 if state 0 occurs whereas
with probability 1 − π = 0.5 income will equal 6 in period 1 if state 1 occurs.
Consumer has to make decision about her consumption and saving for period 0
before uncertainty is resolved.
(i) Write down the Euler equation, and find the expected value and variance of
income in period 1.
(ii) Find the optimal consumption and saving in period 0, and consumption in
period 1 in both states of nature.
1

(iii) Does your answer for the optimal consumption in period 0 and savings differ
from the answer to (2a), and why it does or why it doesn’t? (Hint: recall
saving-for-a-rainy day motive of a PIH consumer.)
(c) Imagine now instead that at date 1 state 0 income is 0 with π = 0.99 and at date
1 state 1 income is 300 with probability 1 − π = 0.01, while income at date 0 still
equals 3. Find the expected value and variance of income at date 1. What would
be the optimal choice of consumption and savings at date 0 and why? (Hint:
recall saving-for-a-rainy day motive of a PIH consumer.)
(d) Assume now that a consumer has log-preferences instead and solve (2b). Comment
on the differences in your results in light of precautionary savings.
3. Consider the Permanent Income Hypothesis we studied in class. Preferences are
quadratic, u(ct) = −
1
2
(ct − c ̄)
2
; planning horizon is infinite and consumer starts her
life at time 0; β(1 + r) = 1; income stream is known as of time 0. Consider individual
X whose income is 0 in even periods (periods 0, 2, 4, 6, etc.) and 190 in odd periods
(periods 1, 3, 5, 7, etc.); and individual Y whose income equals 100 in all periods. The
real interest rate, r, equals 1/9, and 1
1+r = 0.9.

(a) Find the optimal consumption plan for both individuals assuming the credit mar-
ket is perfect.

(b) Find the optimal consumption plan for both individuals assuming they are not
allowed to borrow at all.
4. This problem consists of two parts, I and II.

Part I
Consider the Arrow-Debreu setup with 2 agents, two periods (0 and 1), and two states
of the world in period 1 (labelled 1 and 2). Assume that preferences for each individual
i = 1, 2 are given by the utility function:
U
i = −
1
2
[c
i
0 − c ̄]
2 −
β
2
π(1)[c
i
1
(1) − c ̄]
2 −
β
2
π(2)[c
i
1
(2) − c ̄]
2
.

Assume that π(1) = π(2) = 1 Econ 482 Advanced Macroeconomic Theory II Assignment
2
, β = 0.9. Parameter ̄c is the same for both agents.

In period 0, incomes of agents 1 and 2 are both equal ̄y.
In period 1:
• for agent 1, income will equal ̄y + in state 1, period 1 and ̄y − in period 1 if
state 2 occurs (0 < < y ̄);
• for agent 2, income in period 1, state 1 equals ̄y −, and income in period 1, state
2 equals ̄y + . Econ 482 Advanced Macroeconomic Theory II Assignment

2

AD prices are denoted as q(1) and q(2), where q(s), s = 1, 2 is the price of AD security
at time 0 that delivers of one unit of consumption if state s realizes at date 1, and
nothing otherwise.

(a) Solve for equilibrium prices of Arrow-Debreu securities and the equilibrium con-
sumption allocations for each agent at each date and state.

(b) Given the Arrow-Debreu prices you found above, find the real interest rate for
a riskless asset in the economy that delivers 1 unit of consumption at date 1
regardless of the state of the economy. Econ 482 Advanced Macroeconomic Theory II Assignment
Part II

Assume everything as above with the only difference that the market for contin-
gent claims is shut down and instead there’s a market for a riskfree bond; the

bond costs 1 unit of consumption today and pays off 1 + r units of consumption
tomorrow, regardless of the state, where r =
1
9
. Both agents take the interest rate Econ 482 Advanced Macroeconomic Theory II Assignment

as given when making their consumption decisions.
(c) Find the consumption allocations of each agent at each date and compare them
with the allocations you found in Part I. What setup, in Part I or II, delivers a
higher expected utility? (You don’t need to calculate those utilities if you know
the answer.) Briefly explain why.
5. Consider the perfect credit-market setting and the consumption choice of an individual
who lives for two periods only, with consumption c0 and c1 and incomes y0 and y1.
Suppose that the utility function in each period is

u(c) =  Econ 482 Advanced Macroeconomic Theory II Assignment
ac −
b
2
c
2
, for c < a
b
;

a
2
2b
, for c ≥
a
b
.

Although the above utility function is quadratic, we rule out the possibility that a Econ 482 Advanced Macroeconomic Theory II Assignment
higher consumption level reduces utility.
(a) Plot marginal utility as a function of consumption.
(b) Suppose that β = 1, r = 0, y0 =
a
b
and

y ̃1 =
a
b + σ, with prob. 1
2
;

a
b − σ, with prob. 1
2
.

Solve the problem and discuss the effect of a higher σ on consumption at time 0 Econ 482 Advanced Macroeconomic Theory II Assignment
and saving. Further
6. Provide brief(!) answers to the questions below.
3

(a) Suppose that all individuals have preferences u(w) = −
1
5
exp−5w, you collect the
data on individual wealth and risky asset holdings, and run the regression ait =
β0 + β1wit + eit, where ait is individual i’s holdings of risky assets at time t, wit is
individual i’s total wealth holdings and eit is error term. What would you expect
to obtain for βˆ

1 (estimated β1) in such a regression in the context of the canonical

portfolio theorem?
(b) Suppose an individual lives for 2 periods, has preferences u(c) = − Econ 482 Advanced Macroeconomic Theory II Assignment
1
5w
−5 +
1
5
,

income at date 0 of 100, income at date 1 of 0 with probability 1 Econ 482 Advanced Macroeconomic Theory II Assignment
4
and 1000 with

probability 3
4 Econ 482 Advanced Macroeconomic Theory II Assignment
. What would be that individual’s natural borrowing constraint if

net real interest rate equals 10% and the credit market is perfect?
(c) Assume the Arrow-Debreu world with a complete set of contingent consumption
claims. Assume that there’re 2 dates, 0 and 1, and 2 possible states in period 1,
and the prices of Arrow-Debreu securities are equal 1
2
and 2 for date 1 state 0 and
date 1 state 1, respectively. Find the price of an asset that promises 2 units of
consumption at date 1 if state 0 occurs and 1
4
units of consumption at date 1 if

state 1 occurs.
(d) Suppose that all individuals have preferences u(c) = ln(c), you collect the data on
individual consumption and individual incomes, and run the regression ∆ log cit =
β0+β1∆ log yit+β2∆ log ̄yt+eit, where cit is individual i’s consumption at time t, yit
is individual i’s income, ̄yt

is aggregate income at time t, ∆ log xt = log xt−log xt−1
(where x is consumption or income), and eit is error term. What are β1 and β2
under the null hypothesis of complete insurance markets? Econ 482 Advanced Macroeconomic Theory II Assignment

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