ECON30009/90080 Macroeconomics
James Hansen
Assignment 1
(Due no later than August 25, Friday, 4pm)
In this assignment, you are asked to study modi…ed versions of the life-cycle model and
the growth accounting exercises discussed in the Lectures. You need to type up the solutions
to your assignment (e.g., in Word or LaTeX) and convert your solutions to a PDF …le for
online submission once they are complete. Handwritten submissions are also permitted but
they must be clearly legible so they can be marked.
Please note this is a group assignment (up to 4 students per group). Please submit your
solutions via the LMS. All students within the same group will receive the same mark. No
two groups may submit the same assignment. You can collaborate with members of your
own group (and all group members must provide input), but you may not collaborate across
groups.
Question 1: A Modi…ed Life-Cycle Model with Exogenous Income and Interest
Rates (25 marks in total)
Consider the life-cycle model discussed in Lectures 3-5, but now suppose that each
young individual born in period t has the following preferences:
U (cyt; cot+1) = 2
p
cyt + epco;t+1
where cyt is consumption when young in period t, cot+1 is consumption when old in
period t+1; and e is a discount factor that governs the patience of the young individual
with 0 < e < 1. Assume that individuals received an exogenous endowment of income
yt each period (i.e. yt units of output when young, and yt+1 units of output when old)
where output grows at a constant rate g so that yt+1 = (1 + g) yt. Assume that the
real interest rate is exogenous and falling so that rt+1 = rt where 0 < < 1 is a
parameter with r1 > 0 given.
(a) (5 marks) Write down the period t and period t + 1 budget constraints and use
them to derive the lifetime budget constraint. Explain the intuition for the result
you …nd.
(b) (5 marks) Formulate the individuals’utility maximisation problem and solve it.
Explain how growth rate in income received over the life time g a¤ects the optimal
consumption decisions made by an individual.
(c) (5 marks) Using your solution from (b), show how the parameter a¤ects con-
sumption when young, the optimal level of saving and consumption when old.
Explain the intuition for your result.
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(d) (5 marks) Show graphically, including with income and substitution e¤ects, how a
change in the growth rate in income g and how the persistence of the real interest
rate a¤ect the optimal consumption pro…le of individuals over their lifetime.
(e) (5 marks) Now suppose the individual utility functions are modi…ed so that all
individuals have lifetime preferences
U (cyt; cot+1) = 2
p
cyt + epco;t+1 c0g c1g2
where g is again the exogenous growth rate of income, and c0 > 0 and c1 > 0 are
constants. What might be the interpretation of the additional terms c0g c1g2
in this utility function? Do you think the solutions derived above in parts (a) to
(d) would maximise the welfare of an individual with these preferences. Explain
why or why not.
Question 2: CES Production Function and Growth Accounting
(25 marks in total)
Consider a growth accounting exercise where the production function for aggregate
output (Yt) is now given by
Yt = [ (AtKt)
+ (1 ) (AtLt)]
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where 0 < < 1 and 0 < < 1, At is exogenous productivity, Kt is the aggregate
capital stock and Lt aggregate labour inputs.
(a) (5 marks) Compute the …rst partial derivatives of aggregate output with respect to
capital and labour (i.e., @Yt
@Kt
and @Yt
@Lt
). Does productivity, At, a¤ect the marginal
product of capital and the marginal product of labour in this production function?
Explain.
(b) (5 marks) Show that the marginal rate of technical substitution between capital
and labour, MRTSKt;Lt
@Yt
@Kt
@Yt
@Lt
must be equal to the ratio of factor costs for …rms
producing under perfect competition. That is, show
MRTSKt;Lt =
rt
wt
where rt and wt are the real cost of capital (real interest rate) and the real wage
rate. Explain how an increase in the real wage wt (holding …xed the real interest
rate) a¤ects the aggregate capital to labour ratio in the economy?
(c) (5 marks) Assuming labour hours are held …xed, derive an expression approxi-
mating the growth rate of output per worker as a function of the capital to labour
ratio in periods t and t + 1 and growth in productivity At. That is derive an
expression of the form:
ln
Yt+1
Yt
ln Lt+1
Lt
= f

kt; kt+1; g
A
t

2
where kt KtLt ; kt+1
Kt+1
Lt+1
and gAt ln At+1At and you need to …nd the closed-form
expression representing the function f .
(d) (5 marks) Using your results from (c), calculate the following (linear) approxima-
tion of growth in output per worker
ln
Yt+1
Yt
ln Lt+1
Lt
gA+ @f
@kt jkt=k

kt k

+
@f
@kt+1 jkt+1=k

kt+1 k

+
@f
@gAt jgAt =gA

gAt gA

(1)
where, for example, @f
@kt jkt=k denotes the partial derivative of the function f
kt; kt+1; g
A
t
with respect to kt evaluated at the steady state capital to labour ratio (i.e., where
kt = k) and where gA is the steady state rate growth of productivity (i.e., the
long-run value of gAt ). (Note you do not need to prove that the approximation
(1) exists, you can simply take this approximation as given and use your previous
results in (c) to compute @f
@kt jkt=k;
@f
@kt+1 jkt+1=k
and @f
@gAt jgAt =gA
.)
(e) (5 marks) Using your results in (d), explain what can drive short- and long-run
growth in this economy. Are your results consistent with the growth accounting
exercises we discussed in the lectures when assuming a Cobb-Douglas production
function? Explain why, or why not.
END OF ASSIGNMENT.