Assignment 3 - Production Function Analysis

Assignment 3 Overview

• This assignment focuses on the production function and the dynamics of the capital accumulation model. It specifically examines steady-state conditions and their sensitivity to changes in the saving rate.

Production Function

• The production function is represented by the equation:

  • Y = ext{√}(KL)
    where:

  • Y = Total output

  • K = Capital

  • L = Labor

• The relationship between capital and labor per effective labor is defined as:

  • k = rac{K}{L}

  • Initial condition given is:

  • k = 4

Given Parameters

• Savings rate (s) = 0.1

• Depreciation rate ( ext{δ}) = 0.1

Questions Breakdown

Q1: Finding Steady-State Values

• Objective: Find the steady-state values for capital per effective labor (k), consumption per effective labor (c = C/L), and output per effective labor (y).

Q2: Mathematical Solution for Steady-State

• Steady-state definition: The change in capital per effective labor (Δk) is defined by:

  • Δk = sy - δk = 0

  • This implies that at steady-state, the savings equal the depreciation.

• Rearranging gives:

  • sy = δk

• Solve for each variable as required from the initial conditions.

Q3: Impact of Changing Saving Rates on Steady State

• Initially set saving rate to: s = 0.1

• Objective: Find steady-state values for y, c, and k when saving rate changes to:

  • s = 0.2

• Further Calculation Requirements:

  • Repeat for the following saving rates:

  • s = 0.3

  • s = 0.4

  • s = 0.5

  • s = 0.6

  • s = 0.7

Q4: Optimal Saving Rate for Maximizing Steady-State Consumption

• Objective: Identify the optimal savings rate that maximizes consumption per effective labor (c) at steady state.

• Analyze the relationship between savings rate, output, and consumption in determining optimal conditions for sustained economic growth.