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 onor 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 returndate, 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 notbe 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 notforget to include your name and SID.

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

1PROBLEM 1. (10 Marks) In our study of a simple model of money, we represented 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 end 写 ECOS3010 mathematical equationsowed with yt units of the consumptiongood 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 realmoney

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 anarbitrary 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 Indicators 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 %) andpopulation growth (annual %), and round the final result to four decimal points.] [4marks]

(d) We further breakdown the assumption of the constant stock of fiat money,now we have:Derive the new rate of return on fiat money for Japan over the past decade. Doyou obtain a different result for the value of money in Japan and its implications for

he price level? [Hint: Use the data from 2014 to 2023. For simplicity, employ thearithmetic mean for broad money growth (annual %), and round the final result to

our decimal points.] [2 marks]

2PROBLEM 2. (10 Marks) Let us extend our model from two periods to alife-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 thesecond period, agents enter the labour force and supply labour elastically, receivingwage compensation, which equals to ωl. In the third and final period, agents retire

nd 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 lifetimeutility function is given as:where utility function for consumption and labour supply are:he periodical real interest rate is r. We use a simple notation of real demandfor fiat money (money holdings) from textbook, where qt = vtmt . All parameters are

ssumed to be postive. For your understanding, the first-period budget constraint isAs the central planner, you are concerned about consumption decision for agents andthinking 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. 

(g) What effects does an increase in β have on real money balances and the lifetimeconsumption 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,= 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 doesreal wage affect the labour supply when initial endowments are extremely small, say

y0 → 0? What is the underlying intuition behind this result? [2 marks]