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.