economy with a growing population

INSTRUCTIONS: Review the following very carefully:
This assignment contains 3 questions based on text-book end of the chapter questions featuring concepts from Chapters 1 and 2.
Consider this assignment as a prep work for the upcoming midterm. Therefore, it is required and beneficial for each of you to work on it individually. Any indication of academic dishonesty will result in zero grade for each of the students involved.
SUBMIT YOUR ANSWERS IN THIS DOCUMENT ONLY. ANSWERS SUBMITTED IN ANY OTHER DOCUMENT OR ELSEWHERE WILL NOT BE ACCEPTED FOR GRADING AND THE GRADE FOR THE ASSIGNMENT WILL BE ZERO.
The assignment must be submitted in .doc format or .pdf format. YOU MAY WRITE YOUR ANSWERS USING PEN/PENCIL AND SUBMIT THE SCANNED IMAGES OF YOUR ANSWERS IN THIS DOCUMENT ONLY.
However, the answers must be clearly legible for grading purposes.

QUESTION 1 (Total 5×3 = 15 marks)
Consider an economy with a growing population in which each person is endowed with y1 when young and y2 when old but the endowment in the old age is sufficiently small that everyone wants to consume more that y2 in the second period of life.
Find the feasible set. What is the maximum young consumption possible? What is the maximum old consumption possible?
Assuming stationary equilibrium with no population growth, draw a well labeled graph showing Golden Rule allocation using same assumptions of preferences from Chapter 2. Your graph must reflect the answers you have obtained in part i.
Consider monetary equilibrium now. Find the equation representing the equality of supply and demand in the market for money.
In the monetary setting assuming stationary equilibrium with constant money supply, find the real return for money.
Draw the budget set for a person in this monetary equilibrium in a well labeled graph. Does this monetary equilibrium maximize the utility of future generations? Explain your answer in 4-5 sentences.

QUESTION 2 (Total 3×5 = 15 marks)
In chapter 2, we modeled growth in an economy by a growing population. We could also achieve a growing economy by having and endowment that increases over time. To see this, consider the following economy: Let the number of young people born in each period be constant at N. There is a constant stock of fiat money, M. Each young person born in period t is endowed with yt units of the consumption good when young and nothing when old. The individual endowment grows over time so that yt = 3 yt-1. For simplicity, assume that in each period t, individuals desire to hold real money balances equal to one-third of their endowment, so that vt mt= yt / 3.

Write down equations that represent the constraints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint. Show your steps. (5 marks)
Write down the condition that represents the clearing of the money market in an arbitrary period t. (5 marks)
Find the real rate of return of fiat money in a monetary equilibrium. Show all your steps and explain your answers to avoid grade penalty. (5 marks).

QUESTION 3 (Total: 15 marks)
This is a question from Chapter 2 Appendix section. Suppose in an economy utility function is given by:
ln(c1,t) + b ln(c2,t+1) where, b is a positive parameter. Given the young person’s real demand for money is
(a) Prove that real money balances is given by: q⋆ =by/(1 + b). SHOW YOUR STEPS.
(b) Derive expressions for the lifetime consumption pattern: SHOW YOUR STEPS.
(c)What effects does an increase in b have on real money balances and the lifetime consumption pattern? Give an intuitive interpretation of the parameter b.