Economic Growth: Theory and Empirics

Four stages in economic growth thinking:

  1. 1750-1869: David Hume, Adam Smith, Thomas Malthus, David Ricardo, J.S. Mill and Karl Marx
  2. 1870- 1939: Alfred Marshall, Joseph Shumpeter, Colin Clark, Kuznets, Hoffman, and Roy Harrod
  3. 1940-1985: Solow and Swan
  4. 1986 to present: Paul Romer, Robert Barro, Phillip Aghion, Oded Galor, Daron Acemoglu...

Stylized facts about growth (what reality are you trying to explain):

Important issues:

What does production function capture? Work at home? Is it OK to assume labor and knowledge are exogenously determined? 

Solow Model: 

\(Y (t) = F(K (t), L (t)A (t)\)
Assumptions: constant returns to scale (CRS), and growth rate of L and A is exogenous (something that the model is not intending to explain)
Properties of a production function: has three arguments all of which are function of time, AL ("effective labor" enters multiplicatively, this specification implies that K/Y is constant, homogenous of degree 1 -- constant returns to scale, inputs other than K, A, and L are unimportant (no land or resources, which don't change implications anyway)
\(\therefore\) Cobb-Douglas is convenient and easy to understand 
\(F(\frac{K}{AL}, 1) = \frac{1}{AL} F(K,AL)\)
\(y \equiv \frac{Y}{AL}; k \equiv \frac {K}{AL}\)
\(y = f(k)\)
Remember, \(F(K,L) = K^\alpha (AL^{1-\alpha})\)
\(\therefore f(k) = k^\alpha\)
Evolution of inputs:
\(\dot L (t) = nL(t)\), growth rate of labor (exogenous)
\(\dot A(t) = gA(t)\), growth rate of knowledge (exogenous)
\(\dot K(t) = sY(t) - \delta K(t)\), growth rate of capital (savings) minus depreciation (endogenous-- asking, what drives K?)

Dynamics of the model (embedded all three equations into one):

the model is determined by the movement of k over time
\(\frac{\dot K}{AL} = \frac{sY}{AL} - \frac{\delta K}{AL}\)
\(\frac{\dot K}{AL} = sy - \delta k\)
\(\dot k = \frac{\dot K}{AL}\)
Using chain rule: \(\dot k = \frac{\dot K}{AL} - \frac{K}{AL}(\frac{\dot L A + \dot A L}{AL})\)
Since \(\frac {K}{AL} = k\) and \(\frac{\dot L}{L} = n\) and \(\frac {\dot A}{A} = g\)
\(\dot k + k(n + g) = \frac{\dot K}{AL}\)
\(\dot k = sy - \delta y - nk - gk\)
Or more precisely, \(\dot k(t) = sk^\alpha - (n + g+ \delta)k(t)\)
So, if \(sk^\alpha = (n + g+ \delta)\) then k is not moving (steady state)
Solution:
\(k_t^{1- \alpha} = [k_0^{1- \alpha} - \frac{s}{(n + g+ \delta)}]e^{-(1- \alpha)(n+g+ \alpha)t} + \frac{s}{(n+g+ \delta)}\)
Steady state value of k, k*:
\(k ^\star = [\frac{s}{n+g+\delta}]^\frac{1}{1-\alpha}\)
So, k* is being replenished by savings/investment, then diminished by the growth rates of A (g) and L (n) and depreciation (\(\delta\)). And in steady state, this means that \(\dot k = n+g\) (growth rate of capital per unit of effective labor equals the growth rate of those two factors). 

E.g.: a change in savings rate

Problems with Solow Model

Growth Accounting

Useful for identifying the proximate causes of growth in seeking to find out what fraction of growth is due to increases in factors of production and then from everything else
Works by taking partial derivatives of each component to find the residual:
\(\frac{\dot Y(t)}{Y(t)} - \frac {\dot L(t)}{L(t)} = \alpha_K(t)[\frac{\dot K}{K} - \frac{\dot L}{L}] + R(t)\)
Recent application is by understanding productivity slowdown