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ECOS3010 Assignment
Assignment
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ECOS3010: Assignment 1
PROBLEM 1. (10 Marks) In our study of a simple model of money, we rep-
resented economic growth through a growing population. Recall the market clearing
condition, where the total demand for fiat money must equal the aggregate supply.
This condition implies that:
vt =
Nt(y − c1)
Mt
We have the population dynamic is given as:
Nt+1 = nNt
Each young person born in period t is endowed with yt units of the consumption
good when young and nothing when old. The endowment grows over time so that:
yt+1 = αyt
where α > 1. Assume that in each period t, people desire to hold real money
balances equal to θ of their endowment, where 0 < θ < 1 so that:
vtmt = θyt
There is a constant stock of fiat money, M .
(a) Derive the lifetime budget constraint. [2 marks]
(b) What is the condition that represents the clearing of the money market in an
arbitrary period t? Determine the real return of fiat money in a monetary equilibrium.
How does the percentage of holding endowment affect the real return of fiat money?
[2 marks]
(c) Using the database developed by the World Bank (World Development Indi-
cators Link), find the data for Japan over the past decade to determine the values
for α and n. Assess whether the value of money in Japan is increasing or decreasing.
Briefly Discuss the implications for the price level. [Hint: Use the data from 2014 to
2023. For simplicity, employ the arithmetic mean for GDP growth (annual %) and
population growth (annual %), and round the final result to four decimal points.] [4
marks]
(d) We further breakdown the assumption of the constant stock of fiat money,
now we have:
Mt+1 = zMt
Derive the new rate of return on fiat money for Japan over the past decade. Do
you obtain a different result for the value of money in Japan and its implications for
the price level? [Hint: Use the data from 2014 to 2023. For simplicity, employ the
arithmetic mean for broad money growth (annual %), and round the final result to
four decimal points.] [2 marks]
2
PROBLEM 2. (10 Marks) Let us extend our model from two periods to a
life-cycle economy. Agents are endowed with y0 when they are young. In their youth,
they do not work as they are accumulating skills for the next period. During the
second period, agents enter the labour force and supply labour elastically, receiving
wage compensation, which equals to ωl. In the third and final period, agents retire
and enjoy all the money holdings accumulated from the previous periods. Agents
can save and borrow every period and discount utility at rate β. The agent lifetime
utility function is given as:
U =
3∑
t=1
βt−1u(ct) + βv(l)
where utility function for consumption and labour supply are:
u(ct) = lnct
and
v(l) = ln(1− l)
The periodical real interest rate is r. We use a simple notation of real demand
for fiat money (money holdings) from textbook, where qt = vtmt. All parameters are
assumed to be postive. For your understanding, the first-period budget constraint is
given as:
c1 + q1 ≤ y0
The second-period budget constraint is:
c2 + q2 ≤ (1 + r)q1 + wl
The third-period budget constraint is:
c3 + q3 ≤ (1 + r)q2
and lastly,
q3 = 0
As the central planner, you are concerned about consumption decision for agents and
thinking about the labour supply of the agents.
(e) Based on above constraints, derive the lifetime budget constraint. [1 mark]
(f) Setup the Lagrangian equation to represents the optimisation problem. [1
mark]
(g) What effects does an increase in β have on real money balances and the lifetime
consumption pattern? Give an intuitive interpretation of the parameter of β.[1 mark]
(h) Derive the expressions for the lifetime optimal consumption for first period.
[Hint: You are going to solve the consumption as a function of the given parameters,
i.e. c∗1 = f(yo, ω, r, β). You can start with deriving the FOCs.] [4 marks]
(i) Derive the labour supply at optimal. [1 mark]
(j) How does the initial endowments y0 affect the agent labour supply? How does
real wage affect the labour supply when initial endowments are extremely small, say
y0 → 0? What is the underlying intuition behind this result? [2 marks]