Intermediate Macro CH 6

Chapter 6: Growth

Outline

• Questions:

  • Why are we richer now than 50 years ago?

  • Why are some countries richer than others?

• Solow growth model

• Data on infant mortality rates:

  • 20% in the poorest 1/5 of all countries

  • 0.4% in the richest 1/5

• Historical context: One-fourth of the poorest countries have experienced famines in the past 30 years.

• Economic growth is crucial as it raises living standards and diminishes poverty.

• The chapter seeks to examine:

  • Why some countries experience growth miracles, such as the Asian Tigers.

Economic Growth Goals

• Understand the reasons behind poverty in certain countries.

• Design effective policies to facilitate growth in impoverished nations.

• Analyze how domestic growth rates are influenced by external shocks and governmental policies.

The Importance of Long-term Economic Growth

• Any factor influencing long-term economic growth, even marginally, can have extensive effects on living standards over time.

  • Example statistics demonstrating growth effects:

  • 85.4% increase in standard of living over 25 years due to growth.

  • 243.7% increase over 50 years.

  • 1,081.4% increase over 100 years.

The Solow Model

• Transition from static models (like a snapshot of the economy) to dynamic models (a view of the economy over time).

• Introduced by Robert Solow, this model is pivotal in economic growth studies and serves as a benchmark for modern growth theories.

• Focus on the determinants of long-term economic growth and standards of living.

Differences from Chapter 2’s Model

• Starting point: All chapters begin with the function $Y = F(K,L)$

  • Key Differences:

  • Capital (K) is variable: it increases with investment and decreases with depreciation (machines wear out).

  • Labor (L) is variable: determined by population growth.

• Simplification:

  • Omitted government consumption (G) and taxes (T) for clarity. Simplified model: $Y = C + I$ (where consumption (C) includes government consumption and investment (I) includes government investment).

• Assumed constant savings rate (s):

  • Consumption: $C = (1 - s) Y$

  • Investment: $I = Savings = s Y$

Important Notation

• Lowercase variables denote per capita (or per worker) values of upper case variables:

  • $Y = output$; $y = output ext{ per worker} = Y/L$

  • $K = capital$; $k = capital ext{ per worker} = K/L$

• Note: Savings rate $s$ represents the fraction of income saved; total savings are represented by $S$.

The Production Function

• In aggregate form: $Y = F(K, L)$.

• Define:

  • $y = Y/L = ext{output per worker}$

  • $k = K/L = ext{capital per worker}$

• Assumed constant returns to scale: If z > 0, then $zY = F(zK, zL)$. Assume $z = 1/L$, thus $Y/L = F(K/L, 1)$ leading to $y = F(k, 1)$, and denote it as $y = f(k)$.

The Production Function and Marginal Product of Capital

• $f(k)$ demonstrates diminishing marginal product of capital (MPK):

  • $MPK = f(k + 1) - f(k)$

Growth of Capital (1)

• Capital Dynamics:

  • Invest by adding new capital, but also lose capital when it depreciates (e.g., machines wearing out).

  • Capital depreciation rate is denoted by $ext{rate } ext{δ}$.

  • Intuitively, the change in capital: $riangle K = ext{Investment} - ext{depreciation} = sY - ext{δ}K$.

Growth of Capital (2)

• Dividing both sides by capital $K$ gives:

  • $rac{ riangle K}{K} = s rac{Y}{K} - ext{δ}$

Growth of Labor

• Assume population grows at a constant rate $n$:

  • $rac{ riangle L}{L} = n$

  • There is no distinction between population and labor force for this chapter's purposes.

Model Dynamics: Growth of k

• Key variable: capital per worker $k$.

• Using the formula for growth rates leads to:
  $rac{ riangle k}{ k} = rac{ riangle K}{ K} - rac{ riangle L}{ L}$

• Substituting the found expression gives:
  $rac{ riangle k}{ k} = s rac{Y}{K} - ( ext{δ} + n)$

Key Growth Equation

• The central expression of the Solow model:

  • $rac{ riangle k}{ k} = s f(k) - ( ext{δ} + n)$

The Equation of Motion for k

• Determines capital behavior over time, affecting all other variables (income per person: $y = f(k)$, consumption per person: $c = (1 - s)f(k)$, investment per person: $i = sf(k)$).

• The equation: $riangle k = sf(k) - ( ext{δ} + n)k$

Understanding the Motion Equation

• $riangle k = sf(k) - ( ext{δ+n})k$

  • Represents actual investment per worker, this investment must cover depreciation and equip new workers.

• Breakdown of terms:

  • $ext{δ}k$: capital depreciation per worker.

  • $nk$: captures new workers that require equipment proportional to the current level of capital per worker.

The Steady State

• When investment equals the sum of depreciation and equipment for new workers:

  • If $sf(k) = ( ext{δ} + n)k$, then capital per worker remains constant, leading to $riangle k = 0$.

  • Denote steady state capital stock as $k^*$.

• Graphical representation displays the intersection of investment and depreciation lines.

Transition to Steady State

• The economy tends toward steady state.

  • Case 1: Starts with k_1 < k^*

  • Case 2: Starts with k_1 > k^*

• Both scenarios illustrate the return to the steady state.

Numerical Example

• Consider the production function as $Y = F(K, L) = rac{1}{2} K^{ rac{1}{2}} L^{ rac{1}{2}}$.

• Next, details of the numerical example will illustrate the transition towards steady state.

Predictions Based on Capital Dynamics

• Once steady state is established, deviations in investment illustrate shifts in capital per worker.

• Deviation forecasts may vary based on shocks and policy implementations.

Convergence Evidence from the Solow Model

• Predictions on growth rates and population impacts.

• Conditional convergence outlined: economies converge to their own steady states influenced by individual savings and population growth rates.

Summary of Growth Factors

• Key Insights:

  • Investment strategies and economic policies critically shape the trajectory towards growth.

  • Changes in capital dynamics and human capital development significantly influence overall growth rates.

• Steady-state growth predictions and readiness to embrace new economic models will dictate future growth trajectories in the respective economies.