Economic Growth: Theory and Empirics
Four stages in economic growth thinking:
- 1750-1869: David Hume, Adam Smith, Thomas Malthus, David Ricardo, J.S. Mill and Karl Marx
- 1870- 1939: Alfred Marshall, Joseph Shumpeter, Colin Clark, Kuznets, Hoffman, and Roy Harrod
- 1940-1985: Solow and Swan
- 1986 to present: Paul Romer, Robert Barro, Phillip Aghion, Oded Galor, Daron Acemoglu...
Stylized facts about growth (what reality are you trying to explain):
- Economies grow over time in the long-run
- There are vast differences in the standard of living across countries
- Factors of production (K & L) share of total income are roughly constant: \(F(cK, cAL) = cF(K,AL)\)
- Ratio of capital to labor (K/L) is roughly constant over time (1:3): \(F(K,L) = K^\alpha (AL^{1-\alpha})\)
- Marginal return to capital is +/- constant (mostly true for UK... longest time series)
- Productivity tends to increase over the very long run (history of world GDP, 444 \(\rightarrow\) 6000
- Share of consumption in GDP has remained constant (~70%): \(\dot k (t) = sY(t) - \delta k(t)\)
- Financial development precedes economic development and growth (Hamilton's predictions)
- There are structural transformation in the development process that a generalized production function cannot capture (different industries)
- There are growth miracles and disasters (i.e. China vs. Buenas Aires)
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
- Temporarily increase the growth rate of output per effective worker
- Impact on long-term consumption will depend on whether the new steady-state level of capital is above or below the "Golden Rule" level
- Consumption will be equal to: \(c = y - (n+g+ \delta)k\), or the amount of output that is not saved
- Max consumption by: \(f'(k) = n + g + \delta\), or tangent line of \(f(k)\)
- ^THAT is the Golden Rule
Problems with Solow Model
- Implied differences across countries in capital per worker are extremely large (US to India would be over 100x)
- Implied differences across countries in the marginal return to capital are also implausibly large
- g is exogenous (but improvements on current capital are the bulk of growth)
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