[SOLVED] ECOS3010 Assignment

ECOS3010 Assignment

Problem 1

(a) Lifetime Budget Constraint Derivation

Given:

• Population growth: Nt+1 = nNt

• Endowment growth: yt+1 = αyt

• Money demand: vtmt = θyt

Young Period (t):

Old Period (t+1):

Intertemporal Budget Constraint:

Final Lifetime Budget Constraint:

Problem 1

(b) Money Market Clearing and Real Return on Fiat Money

(i) Money Market Clearing Condition

In period t, the money market clears when total real money demand equals total real money supply:

where:

• Ntθyt = Aggregate real money demand (from young generation)

• vtM = Real value of total money supply

(ii) Real Return on Fiat Money

The real return on fiat money () is derived as follows:

(iii) Effect of θ on Real Return

• θ cancels out in the return calculation, making the real return independent of money demand preferences.

• Economic intuition: While θ affects the level of money demand (vt), it does not influence its growth rate ().

Problem 1

(c) Empirical Analysis for Malaysia

Problem 1

(d)

Problem 2

(a)

Given the four-period budget constraints:

Express q1 from Period 1

q1 = y0 − c1

Substitute into Period 2

Substitute into Period 3

Substitute into Period 4

Lifetime Budget Constraint

The agent’s lifetime budget constraint is:

Problem 2

(b)

The Lagrangian for the agent’s optimization problem is:

where R = 1 + r.

Problem 2

(c)

First-Order Conditions

Maximize the Lagrangian:

Solve Using Budget Constraint

Substitute into budget constraint:

Final Answer:

Problem 2

(d) Optimal Labor Supply

Given from Problem 2(c):

The optimal first-period consumption is:

First-Order Conditions for Labor:

Solving the System:

Substitute and into :

Multiply through by Γ where Γ = 1 + β + β2 + β3:

This yields the quadratic equation:

Final Answer: Optimal Labor Supply

Problem 2

(e)

An increase in the initial endowment y0 leads to:

1. Higher initial consumption

2. Lower optimal labor supply in both periods: ↓ and ↓

Intuition: Richer agents substitute labor for leisure, reducing their need to work for additional income.