Neo-classical Growth Model: Capital Accumulation, Savings, and the Steady State

Definition and Focus: Neoclassical growth theory primarily focuses on the process of capital accumulation and its explicit link to the savings decisions of an economy.
Lecture Objectives: * To explain the per capita production function and the fundamental role played by diminishing marginal returns to capital. * To investigate the specific effects of investment, saving, and population growth on long-term living standards and economic growth. * To understand the concept of the "steady state" and its implications for economic growth.
Required Reading Materials: * D, F & S Ch.3 (13th ed): pp. 63-68. * D, F & S Ch. 3 (12th ed): pp. 61-66. * D, F & S Ch. 3 (11th ed): pp. 61-66.
The Framework for Presenting Growth Theory: * Step 1 (Session 3): Examine the economic variables that determine the economy’s steady state. * Step 2 (Session 4): Study the transition of the economy from its current position to the eventual steady state. * Step 3 (Sessions 5 & 6): Integrate technological progress into the model to understand its impact on growth.

Mathematical Expression: The function is represented as: * $y = f(k)$ * In this formula, $k$ represents the capital-to-labor ratio.
Shape and Properties of the Production Function: * Positive Marginal Product: As capital increases, output increases. This demonstrates that more machines or equipment leads to more production. * Diminishing Marginal Product of Capital: While each additional unit of capital adds to total production, it adds less than the previous unit. * Rate of Increase: Consequently, output per capita increases at a decreasing rate. Graphically, the production function is concave to the horizontal axis.

Required Investment: The level of investment necessary to maintain a constant capital-to-labor ratio ( $k$ ) is dependent on two primary variables: the population growth rate and the depreciation rate. * Population Growth ( $n$ ): It is assumed that the population grows at a constant rate, denoted as $n$ . To accommodate new workers entering the labor force, the economy requires an investment of $nk$ to provide them with the same amount of capital tools as existing workers. * Depreciation ( $d$ ): It is assumed that capital wears out at a constant rate, denoted as $d$ . To replace this worn-out capital, the economy requires an investment of $dk$ . * Total Required Investment Formula: The total investment needed to keep $k$ constant is the sum of investment for new workers and investment for depreciation: * $\text{Required Investment} = (n + d)k$
Savings Formulation: * The model assumes that savings is a constant function of income, denoted by the savings rate $s$ . * Per Capita Savings: This is expressed as $sy$ . * Substitution: Given that income equals production ( $y = f(k)$ ), per capita savings can be rewritten as: * $\text{Per Capita Savings} = sf(k)$

Definition of Steady State: An economy reaches a steady state when per capita income ( $y$ ) and the capital-to-labor ratio ( $k$ ) remain constant over time.
Equilibrium Condition: The steady state is arrived at when the savings generated by the economy are exactly equal to the investment required to provide capital for new workers and replace worn-out machinery. * Formula: $sf(k) = (n + d)k$
Graphic Interpretation: * The Investment Requirement Line: The straight line representing $(n + d)k$ shows the investment required at every level of $k$ to maintain that ratio. * Point D: This represents the point where savings and required investment balance, defining the steady-state capital level. * Adjustment Process (Point k0): Consider an initial point $k_0$ where the capital-output ratio is below the steady state. * If saving > investment requirement: This means there is surplus capital being generated beyond what is needed to maintain the current ratio. As a result, $k$ increases. * The economy moves to the right along the horizontal axis over time. * Point C: This is the designated steady state. At this point, the adjustment process ceases because actual investment equals required investment. The capital-labor ratio becomes stationary, neither rising nor falling.

Convergence Hypothesis: The model implies that countries with identical savings rates, population growth rates, and technological levels should converge toward the same levels of income over time, regardless of their starting positions.
Effect of the Savings Rate: It is a critical distinction that the steady-state rate of growth is not affected by the savings rate. While a higher savings rate can lead to a higher level of steady-state income, it does not change the growth rate once the steady state is achieved.
Growth at the Steady State: * In the steady state, both per capita output ( $y$ ) and the capital-to-labor ratio ( $k$ ) are constant (growth rate is zero). * However, aggregate income continues to grow. At the steady state, aggregate income grows at the exact same rate as the population growth ( $n$ ).