Economic Growth: Solow-Swan Model Overview

References

• Solow-Swan Model notes are based on various resources: Economic Growth (2nd edition), MIT Press (Barro & Sala i Martin, 2004)
• The Facts of Economic Growth (Jones, 2016), Handbook of Macroeconomics, 2A
• Advanced Macroeconomics (5th edition), McGraw Hill (Romer, 2018)

Outline

• Introduction
• Solow-Swan model assumptions
• Dynamics of k
• Steady state
• Level effects
• Golden rule
• Convergence
• Micro-foundations
• Appendix

Introduction

• Figure 26 displays the lack of worldwide convergence in GDP per capita between 1960-2011

The Solow-Swan Model

• Very influential theory of economic growth, simple dynamic model.
• Current income equals current production, based on current capital and labour endowments, and technology.
• Part of current income is saved and invested into additional capital
• Tomorrow's income results from tomorrow's capital and labour, and state of technology.
• Explains how saving, investment, technology, and technical progress affect economic growth.
• Provides empirical predictions to compare with data.

Assumptions

• Agents sell 1 unit of labour services, own and rent capital.
• Income used for consumption and saving (investment in additional capital).
• One final good, output Y, with constant price P = 1.
• Firms hire labor and rent capital using a production function Y = F(K, AL) (AL is "effective" labor, A is labour-augmenting productivity).
• Markets are perfectly competitive.
• Constant returns to scale (CRS).

Assumptions: Neoclassical Production Function

• Constant returns to scale (CRS).
• Positive, diminishing marginal products.
• Inada conditions: lim Fk = ∞ as K approaches 0, lim Fk = 0 as K approaches ∞. Same for FL.
• Production function in intensive form: f(k) = F(K, AL)/AL = F(K/AL, 1). k = K/AL (capital per effective labor), y = Y/AL = f(k) (output per effective labor).

Assumptions: Diminishing Marginal Productivity

• f'(k) > 0: an increase in k leads to an increase in f(k).
• f"(k) < 0: the higher k, the smaller the increases in f(k) when k rises, leading to decreasing returns to capital.

Assumptions: Dynamic Behavior

• Variables are functions of time (t). Dynamic behavior of variables like labor (L), productivity (A), and capital (K) is assumed, accounting for factors like population growth (n), technological progress (g), and depreciation (δ).

Dynamics of k

• k̇ = sf(k) - (n + g + δ)k.
• sf(k) = gross investment per unit of effective labor
• (n + g + δ)k = break-even investment per unit of effective labor

Steady State

• sf(k*) = (n + g + δ)k*. Unique steady state with k* > 0.
• y* = f(k*). Steady state where all variables grow at constant rates.
• k = 0 is ignored.
• K/L and Y/L grow at constant rate g+n.

Steady State & Phase Diagram

• k < k* --> sf(k) > (n+g+δ)k (investment more than break-even).
• k > k* --> sf(k) < (n+g+δ)k (investment less than break-even).

Steady State

• The economy converges to a unique steady state from any starting point k(0) > 0

Golden Rule

• sf(kGR) = (n+g+δ)kGR
• CGR= max{f(k) - (n+g+δ)k} where CGR is maximum consumption.
• This represents the optimal capital stock for maximum long-run consumption.

Golden Rule

• c* = f(k*) - (n+g+δ)k* The optimal saving rate leads to max consumption.
• If S is below SGR, the economy tends to have dynamic inefficiency or over-saving.
• If S is above SGR, it leads to higher initial consumption and lower future consumption.

Absolute Convergence

• identical parameter values lead to the same k*.
• Countries with lower initial capital stocks grow faster toward the common steady state.

Conditional Convergence

• Differences in parameter values lead to different k*.
• Lower initial capital stocks don't necessarily grow faster.

Convergence

• Absolute convergence: converging to same steady state.
• Conditional convergence: differing parameter values could prevent convergence.

Micro-foundations

• Production with side effects (pollution). Is this relevant in the Solow model? What should the government do?
• Model predicts the effects of (1) Income tax reduction on saving rates and (2) News on future productivity levels on today's economic variables.
• Limitations: lacks welfare evaluation and behavioral responses to environment, policies, etc.
• Models like Overlapping Generations and Ramsey address these challenges.

Appendix: Do Differences in Growth Rates Matter?

• Examples showing that even small differences in growth rates lead to large differences in income after years.

Appendix: Lucas on Growth

• Discusses the importance of considering factors like actions governments can take to grow countries like India and Indonesia. The welfare implications are also noted.

Appendix: Solow on Economic Theory

• Theory depends on assumptions that are not always true.
• The art of theorizing is making assumptions that make final results not strongly dependent on the assumptions.
• Crucial assumptions should be realistic.

Appendix: Neoclassical Production Function

• From definitions: Y = F(K, AL), AL = f(k), KF(K, AL) = f'(k) > 0, K^2F(K, AL) = f"(k) < 0.
• Under CRS and competitive markets, inputs are paid marginal products and factor payments exhaust output (wL + rK = Y).

Appendix: Conditional Convergence

• Shows graph of convergence of personal income across Japanese prefectures against the log of 1930 per capita income for growth rates from 1930-1990.