ECOS3010 mathematical equations

ECOS3010: Assignment 1 (Total: 20 marks)

Due 11:59 pm, Friday Aug 30, 2024

1. Homework must be turned in on the day it is due. Work not submitted on or before the due date is subject to a penalty of 5% per calendar day late. If work is submitted more than 10 days after the due date, or is submitted after the return date, the mark will be zero. Each assignment is worth 10% of total weight.

2. TYPE your work (including all mathematical equations). Homework must be submitted as a typed PDF file, with no exceptions. Untyped work will not be graded and will receive a mark of zero. If any question requests a graph, you may draw the graph by hand, scan it, and include it as a figure in the PDF. Please do not forget to include your name and SID.

3. Ensure that working process is clearly articulated, demonstrating your under-standing and methodology. Detailed and logical presentation of the process is crucial and helpful for solving the problem and earning full credit.

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:

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]

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:

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]