ECONOMICS 401 ASSIGNMENT TWO – Due 23 Oct 4pm Instructions

ECONOMICS 401
ASSIGNMENT TWO – Due 23 Oct 4pm
Instructions
◦ All questions should be answered.
◦ Neatness, legibility, style, and grammar all matter.
◦ You are welcome to discuss the assignment and your answers with other students; however,
each student must submit their own assignment.
◦ Instances of blatant plagiarism will be marked zero. I reserve the right to examine students in
person concerning their understanding of their submitted work.
Questions
1. The period-t utility function ut is for a representative consumer,
u(ct , 1 − `t ) = ln ct +
b
(1 − `t )1−γ
1−γ
(1)
with b > 0 and γ > 0. ct is the consumption good, `t is work effort so 1 − `t is leisure. (Assuming
one unit of time for work or leisure available each period.)
(a) Suppose there is only one consumer that lives for just one period and has no initial wealth.
The consumer’s budget constraint is (price of consumption good is normalised to one),
c = w`
The consumer’s optimization problem is,
max
c,`
subject to
b
(1 − `)1−γ
1−γ
c = w`
ln c +
(i) Show that labour choice does not depend upon the wage rate.
(2)
(ii) In a diagram with c on the vertical axis and 1 − ` (leisure) on the horizontal axis, show
the effect of an increase in w and identify the income and substitution effect for these
preferences.
(5)
(iii) Show the effect on labour supply if there is a lump sum tax introduced of τ ; that is the
consumer’s budget constraint becomes c = w`−τ . For simplicity, assume that preferences
are now ln c + b ln(1 − `). Illustrate this effect in a labour market diagram (i.e. show how
the labour supply curve shifts). This relates to the effect of an increase in government
spending and hence taxes.
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Econ 401 Autumn 2014 Assignment Two Page: 2
(b) Suppose the consumer now lives for two periods, can borrow or lend at a known interest rate
r and has the following budget constraints:
c1 + b1 = w1 `1
c2 = w2 `2 + (1 + r)b1
where b1 is lending in period one (borrowing if negative). There is no uncertainty. Lifetime
utility is now,
U = u(c1 , 1 − `1 ) + βu(c2 , 1 − `2 )
with u(ct , 1 − `t ) as defined in (1).
(i) Show that the lifetime budget constraint is:
c1 +
1
1
c2 = w1 `1 +
w2 `2
1+r
1+r
(4)
(ii) The consumer’s problem is to choose {c1 , c2 , `1 , `2 } to maximize lifetime utility subject
to the lifetime budget constraint. Use the first order conditions to show the following
condition holds:
γ1
w2
1 − `1
=
1 − `2
β(1 + r)w1
(4)
(iii) Define the following:
`∗ ≡ (1 − `1 )/(1 − `2 ) w∗ ≡ w2 /w1
so the above becomes,
w∗
` =
β(1 + r)
∗
γ1
Show that the intertemporal elasticity of substitution for leisure is 1/γ; that is,show
∂`∗ w∗
1
=
∂w∗ `∗
γ
(5)
(iv) Suppose microeconomic empirical evidence says that γ is very large. What implication
does this have for the effect of productivity shocks on employment in a real business cycle
model with these preferences?
Background & Reading
The purpose of this question is to work through the conditions that underlie the intertemporal
substitution of labour (leisure) in real business cycle models. It also demonstrates that preference
parameters, often difficult to estimate or judge, are influential. It is adapted from Romer (2001,
p213), Advanced Macroeconomics, 2nd edition.
To answer this question thoroughly and well, you should read chapter 4 in Romer (2006), Advanced
Macroeconomics, 3rd edition, which provides general discussion concerning real business cycle models.
Other supportive and not so supportive readings for real business cycle models are:
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Econ 401 Autumn 2014 Assignment Two Page: 3
* Prescott, E.C. (1986), “Theory ahead of business cycle measurement,” Federal Reserve Bank
of Minneapolis Quarterly Review.
* Summers, L.H. (1986), “Some skeptical observations on real business cycle theory,” Federal
Reserve Bank of Minneapolis Quarterly Review.
2. Consider an infinite horizon small open economy model with no uncertainty. The representative
agent has preferences,
∞
X
Ut =
β s−t u(cs )
s=t
and can borrow and lend at a given real interest rate (gross), 1 + r. The flow constraint for each
period s ≥ t is
cs + bs+1 = ys + (1 + r)bs
where bt is given and the notation is standard. Assume that β = 1/(1 + r). We define the market
discount factor R = 1/(1 + r).
(a) Show that the flow constraints imply the following intertemporal budget constraint:
∞
X
s=t
Rs−t cs + lim RT bt+T +1 =
T →∞
∞
X
Rs−t ys + (1 + r)bt
s=t
(3)
For parts (b)–(d), assume that limT →∞ RT bt+T +1 = 0.
(b) The consumer’s optimization problem is to choose {cs }∞
s=t to maximize
Ut =
∞
X
β s−t u(cs )
s=t
subject to
∞
∞
X
X
Rs−t cs =
Rs−t ys + (1 + r)bt
s=t
s=t
Construct the Lagrangian and find the first order conditions that describe the path for consumption.
(5)
(c) Using the path for consumption and the intertemporal budget constraint, solve for ct ; that is,
show:
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ct
=
∞
∞
i
X
r h
r X s−t
(1 + r)bt +
Rs−t ys = rbt +
R ys
1+r
1 + r s=t
s=t
Provide an interpretation of this solution.
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Econ 401 Autumn 2014 Assignment Two Page: 4
(d) Using your result in part (d), answer the following:
(i) Find an expression for the effect on current consumption from a temporary change in
dct
. Interpret.
current income; that is, find dy
t
(2)
(ii) Find an expression for a permanent increase in income; that is, let dys = dy for all s ≥ t.
t
Solve for dc
dy . Interpret.
(3)
(e) Here we return to the terminal condition limT →∞ RT bt+T +1 = 0.
(i) Consider an individual who begins life with a debt of 100, bt = −100 and constant income,
ys = y, ∀s ≥ t. In each period, they will borrow a further amount to repay the interest
and they will not pay off the original principal. This implies bs+1 = bs + rbs ∀s ≥ t.
Using the model above, find an expression for cs , ∀s ≥ t.
(2)
T
(ii) Find an expression for limT →∞ R bt+T +1 ; specifically, show that this expression is not
zero. What does it imply about the present discounted value of consumption for the
consumer.
(3)
(iii) Now consider a new individual with the same general problem as above. This consumer’s
consumption path is given by the optimal solution for consumption from above:
ct
= rbt +
∞
r X s−t
R ys ,
1 + r s=t
∀t
This consumer is blessed with output that grows over time but at a rate less than the
discount rate r; that is,
yt+1 = (1 + γ)yt , γ < r
Use this to find a solution for ys as a function of yt (i.e. yt is our initial condition and is
given).
(iv) With the solution to part (iii), find a solution for consumption, ct .
(3)
(3)
(v) Now, here is the messy part. Assume bt = 0; find an expression for bt+T +1 ; specifically,
show
i
γ h
bt+T +1 = −
1 + (1 + γ) + (1 + γ)2 + . . . + (1 + γ)T yt
r−γ
As time passes, what is happening to the consumer’s debt?
(5)
T
(vi) Can we use the solution in part (v) to argue that limT →∞ R bt+T +1 = 0? Explain.
(3)
(vii) (Bonus.) For yt given and bt = 0, plot cs , bs+1 , and ys for all s ≥ t.
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Econ 401 Autumn 2014 Assignment Two Page: 5
3. (Based on Romer (2006, pp 218, Question 4.8 & 4.9.) This is a simple dynamic model of
consumption and capital accumulation; the goal here is to find a description of the solution, which
is a very standard intertemporal model. Along the way, we will make use of a solution method often
used in rational expectations models, Method of Undetermined Coefficients.
The economy consists of a constant population of infinitely lived individuals. The representative
individual has an expected utility function
U
= E0
∞
X
t=0
u(ct )
(
1 t
) · u(ct )
1+ρ
= ct − θ(ct + vt )2
where ρ, θ > 0, consumption is such that u0 (c) > 0, ct is consumption per person, and vt is a captures
changes in consumer’s tastes over time. We assume that the shock is persistent, following an AR(1)
process:
vt = ψvt−1 + µt
where −1 < ψ < 1 and µt are mean zero independently and identically distributed ‘taste’ or
‘preference’ shocks.
The production function and the resource constraint for the economy are:
yt
=
akt + et
kt+1
=
kt + yt − ct
Assume that the interest rate, equal to the marginal product of capital a, is equal to ρ. And the
disturbance follows the process:
et = φet−1 + t
with −1 < φ < 1 and t are mean zero independently and identically distributed shocks. These are
‘technology’ shocks. By assumption, technology shocks and preference shocks are uncorrelated with
each other: Et µt t = 0.
Our standard Euler condition for consumption is:
u0 (ct )
= Et (1 + a) ×
1
u0 (ct+1 )
1+ρ
(a) Using the functional form for u(c) and the assumptions above, find a solution for the path of
expected consumption.
(3)
(b) To solve the model, we will guess a general linear form for consumption as a function of the
state variables, kt , vt , and et :
ct = α + βkt + δvt + γet
Based on this guess, provide a solution for kt+1 as a function of the state variables, kt , vt , and
et .
(c) What values must the parameters α, β, δ, and γ have for the first order condition in part (a)
to be satisfied for all values of kt and et ? Having determined these values, write down the
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Econ 401 Autumn 2014 Assignment Two Page: 6
system of dynamic equations for yt , ct , and kt+1 as functions of the state variables kt , vt , and
et .
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You are welcome to check your answer with me at this point as the rest of the question depends
upon getting this part correct.
(d) Conditional on kt , find an expression for dct /dµt and for dct /dt (these are the impact or
instantaneous effects of the shocks). Provide intuition for each of these responses; as well,
explain how the persistence of the shocks governed by ψ and φ affects these responses.
(8)
(e) Solve for the effects of a one time shock to µt for the paths of y, k, and c. To make this
simple, assume (without loss of generality) that dµt = a + 1 − ψ. Specifically, you should find
general expressions for dkt+s , dyt+s , and dct+s as well as determining the limit effects (i.e. as
s → ∞). Note that since kt is given by assumption, dkt = 0; however, dyt and dct are not as
these variables do respond to the shock. (Recall part (d).)
Hints: to proceed, first find expressions for dvt , dvt+1 , dvt+2 and so on. Then work out the
effects for dkt+1 , dkt+2 , and so on until you can generalize to dkt+s . Proceed in the same way
for output and then consumption to obtain expressions for dyt+s and dct+s .
The shock we are considering is temporary in the sense it dies out over time (the effect on vt+s
fades away to zero). But does it have temporary or permanent effects on our model variables
c, y, and k? If permanent, why does this occur?
(f) Provide a plot of the effects of this shock for each of these variables. To be clear, the figure
should be drawn in the type of diagram below.
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[40]
dct+s , dkt+1+s , dyt+s
6
0
s
Total marks: 110