Problem Set 2
Advanced Macroeconomics I Part 2
Contents
Problem 1
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Problem 2
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Problem 3
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Further points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Problem 1
True, False, Uncertain?
1. The lower the net wealth of the household, the more likely it is that the household
is subject to binding borrowing constraints.
Uncertain: Net wealth can be low because the household has a low demand for net
wealth. Net wealth (assets minus liabilities) can also be low when the household is
successful in securing many loans, thus making liabilities large.
2. Households with positive wealth definitely face no binding borrowing constraints
False: For instance, they may be saving for a down payment. Or they may have positive assets but they may still be consuming less than they would in the absence of
borrowing constraints, just because they are afraid these borrowing constraints may
bind in the future.
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3. Risk aversion plays no role in the decision of whether to hold stocks or not
True: This is so in the standard model without any frictions. Uncertain: In the presence of fixed entry costs, risk aversion plays a role because it affects the amount that
the household would put in stocks if it were to gain access to the stock market.
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Problem 2
Quadratic Utility, Precautionary Savings and Borrowing Constraints
Consider the following problem:
max∞ E0
{At+1 }t=0
∞
X
β t U (Ct )
t=0
s.t.
Ct + At+1 = At + Yt
Ct ≥ 0 ∀t
A0 given
1. To show: First Order Conditions imply Et [Ct+1 ] = b1 + b2 Ct
• Either set up the Lagrangian or the Bellman Equation and take the first derivatives.
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• Euler Equation:
U 0 (Ct ) = βEt [U 0 (Ct+1 )]
• with quadratic preferences
α1 − α2 Ct = βEt [α1 − α2 Ct+1 ]
1
α1 (β − 1)
+ Ct
⇔ Et [Ct+1 ] =
βα2
β
⇔ Et [Ct+1 ] = b1 + b2 Ct
2. For β = 1 this amounts to
Et [Ct+1 ] = Ct
⇒ no precautionary savings. Why ?
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Now suppose the household faces the constraint At+1 ≥ 0.
max∞ E0
{At+1 }t=0
∞
X
β t U (Ct )
t=0
s.t.
Ct + At+1 = At + Yt
At+1 ≥ 0
Ct ≥ 0 ∀t
A0 given
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3. Optimality Conditions:
• Lagrangian:
L = E0
∞
X
h
i
β t U (Ct ) − λt (Ct + At+1 − At − Yt ) + µt At+1
t=0
• First Order Conditions plus Complementary Slackness Conditions:
U 0 (Ct ) = βEt [U 0 (Ct+1 )] + µt
At+1 µt = 0
At+1 ≥ 0
µt ≥ 0
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4. To show: U 0 (Ct ) = max{U 0 (Yt + At ), βEt [U 0 (Ct+1 )]}
• The budget constraint is given by Ct = Yt + At − At+1 .
• Because At+1 ≥ 0 we know that Ct ≤ Yt + At .
• If At+1 = 0: Ct = Yt + At and U 0 (Ct ) = U 0 (Yt + At ).
• If At+1 > 0: Ct < Yt + At and U 0 (Yt + At ) < U 0 (Ct ) = βEt [U 0 (Ct+1 )]
• Hence, U 0 (Ct ) = max{U 0 (Yt + At ), βEt [U 0 (Ct+1 )]}
5. With quadratic utility we get
α1 − α2 Ct = max{α1 − α2 (Yt + At ), Et [α1 − α2 Ct+1 ]}
⇔ Ct = min{Yt + At , Et [Ct+1 ]}
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(a) Suppose that at time t the liquidity constraint is not binding while in t + 1 the realization of the income shock makes the household borrowing constrained. How are current
consumption choices affected?
• We know: Et [Ct+1 ] = Et [min{Yt+1 + At+1 , Et+1 [Ct+2 ]}], hence,
Ct = min{Yt + At , Et [Ct+1 ]}
= min{Yt + At , Et [min{Yt+1 + At+1 , Et+1 [Ct+2 ]}]}
• First suppose that in t + 1 the borrowing constraint is not binding: Ct = Et [Ct+1 ] =
Et [Ct+2 ] (using law of iterated expectations).
• But: in t + 1 constraint is binding, hence: Ct = Et [Ct+1 ] < Et [Ct+2 ].
• Current consumption choices are affected by future potential borrowing constraints!
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(b) Suppose that the variance of future income Yt+1 increases.
• Low realizations of income become more likely.
• The borrowing constraint is more often binding:
Et [min{Yt+1 + At+1 , Et+1 [Ct+2 ]}] declines and so does Ct .
• Due to higher income uncertainty savings increase because agents are aware of their
inability to smooth low income shocks via borrowing.
• Precautionary savings as a result of risk aversion and liquidity constraints. No prudence!
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Problem 3
Stockholding
Consider the following problem:
max
∞
{St }∞
t=0 ,{Bt }t=0
E0
∞
X
β t U (Ct )
t=0
s.t.
Ct + Bt + St = St−1 R̃t + Bt−1 Rf + Yt
S0 , B0 given
Ct ≥ 0 ∀t
1. First Order Conditions using the Lagrangian:
• Set up the Lagrangian
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• Take first derivatives with respect to Ct , Bt and St .
• Substituting the Lagrange multiplier yields:
U 0 (Ct ) = βEt [U 0 (Ct+1 )R̃t+1 ]
U 0 (Ct ) = βEt [U 0 (Ct+1 )Rf ]
2. First Order Conditions using Bellman Equation
• Define real cash on hand such that the constraint can be written as
(1)
Ct + Bt + St = Xt
f
Xt+1 = St R̃t+1 + Bt R + Yt+1
(2)
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• Bellman Equation:
V (Xt , Yt ) = max{U (Ct ) + βEt [V (Xt+1 , Yt+1 )]}
subject to (1) and (2).
• First Order Conditions for St and Bt , respectively:
U 0 (Ct ) = βEt V 0 (Xt+1 , Yt+1 )R̃t+1
U 0 (Ct ) = βEt V 0 (Xt+1 , Yt+1 )Rf
• Benveniste-Scheinkman:
V 0 (Xt , Yt ) = U 0 (Ct )
• Combining yields:
U 0 (Ct ) = βEt [U 0 (Ct+1 )R̃t+1 ]
U 0 (Ct ) = βEt [U 0 (Ct+1 )Rf ]
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3. To show: No Zero Stockholding
• Subtract the two first order conditions:
Et [U 0 (Ct+1 )(R̃t+1 − Rf )] = 0
(3)
• Suppose zero stockholding and no correlation between income and stock returns:
Et [U 0 (Ct+1 )]Et (R̃t+1 − Rf ) = 0
(4)
Since U 0 (Ct+1 ) > 0 this is a contraction if there is a premium!
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4. To show: No Zero Stockholding if At = St + Bt ≥ 0.
• The first order conditions become
U 0 (Ct ) = βEt [U 0 (Ct+1 )R̃t+1 ] + µt
U 0 (Ct ) = βEt [U 0 (Ct+1 )Rf ] + µt
where µt is the Lagrange multiplier on the additional constraint.
• Subtracting the first order conditions yields
Et [U 0 (Ct+1 )(R̃t+1 − Rf )] = 0
This is equation (3) because the Lagrange multiplier is the same in both FOCs and therefore cancels.
5. To show: Can Zero Stockholding be ruled out if At ≥ 0 and Bt ≥ 0?
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• The first order conditions are
U 0 (Ct ) = βEt [U 0 (Ct+1 )R̃t+1 ] + µS,t
U 0 (Ct ) = βEt [U 0 (Ct+1 )Rf ] + µB,t
where µS,t and µB,t are the Lagrange multipliers.
• Subtracting the first order conditions yields:
βEt [U 0 (Ct+1 )(R̃t+1 − Rf )] = µB,t − µS,t
• Assume that zero stockholding is the optimum and no correlation between earnings and
stock returns.
βEt [U 0 (Ct+1 )]Et (R̃t+1 − Rf ) = µB,t − µS,t
This is not a contradiction, hence, zero stockholding cannot be ruled out!
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Further points
1. Problem Set 1 - Problem 2: Stochastic income
e−αCt = Et [e−αCt+1 ]
= eEt [Ct+1 ]−
⇔ Ct = Et [Ct+1 ] +
α2 σ 2
2
ασ 2
2
2. Lecture Notes: Structuring Ideas
• model assumptions
• what does the data say ?
• model conclusions
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Empirical justification
Lectures 2, 3, 4
Model structure
Lecture 1
Solution Methods
Problem Sets
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Lagrangian
Bellman equation
Envelope theorem
Guess-and-verify
Function iterations
Numerical solution methods
Consumption-savings
decisions under
uncertainty
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utility functions
expectation formation
inter-temporal discounting
internal or external habits
liquidity constraints
borrowing constraints
• consumption smoothing
• precautionary savings
Empirical Tests
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