Review: The Solow Model


 

Review: Solow Model Key Equations

(1)    $ \left(\frac{Y}{L}\right) = \left(\frac{K}{Y}\right)^\left(\frac{\alpha}{1-\alpha}\right)E $ :: production function

(2)    $ \left(\frac{Y}{L}\right)^{*} = \left(\frac{s}{n+g+\delta}\right)^ \left(\frac{\alpha}{1-\alpha}\right)E $ :: equilibrium balanced-growth path

(3)    $ \frac{d\left(\frac{K}{Y}\right)}{dt} = -(1-\alpha)(n+g+\delta)\left(\frac{K}{Y} - \frac{s}{n+g+\delta}\right)$ :: convergence differential equation

(4)    $ \frac{K_t}{Y_t} = \frac{s}{n+g+\delta} + \left(\frac{K}{Y} - \frac{s}{n+g+\delta}\right) e^{-(1-\alpha)(n+g+\delta)t} $


 

Production Function

The most useful form of the production function:

(1)    $ \left(\frac{Y}{L}\right) = \left(\frac{K}{Y}\right)^\left(\frac{\alpha}{1-\alpha}\right)E $

  • Current output per worker Y/L as a function of:
    • Current capital-output ratio K/Y
    • Production function parameter α
    • Current level of the efficiency of labor E
  • Why most useful? Because it ties directly into the state variable most directly and simply connected to equilibrium: (K/Y)*

 

Equilibrium Balanced-Growth Path

The most useful form of equilibrium:

(2)    $ \left(\frac{Y}{L}\right)^{*} = \left(\frac{s}{n+g+\delta}\right)^ \left(\frac{\alpha}{1-\alpha}\right)E $

  • Why most useful? Because it tells you toward what path the key outcome variable Y/L is going:
    • Tells us in the long run balanced-growth path higher when s/(n+g+δ) higher—and by how much it is higher
    • Tells us in the long run balanced-growth path higher when α is higher—and by how much it is higher
    • Tells us output per worker Y/L along the balanced growth path grows at the same rate E grows—which is at rate g…

 

Convergence Differential Equation

(3)    $ \frac{d\left(\frac{K}{Y}\right)}{dt} = -(1-\alpha)(n+g+\delta)\left(\frac{K}{Y} - \frac{s}{n+g+\delta}\right)$

  • The capital-output ratio K/Y is closing a fraction (1-α)(n+g+δ) of the gap between its current and its balanced-growth steady-state value every year…

 

Integrated Out

(4)    $ \frac{K_t}{Y_t} = \frac{s}{n+g+\delta} + \left(\frac{K}{Y} - \frac{s}{n+g+\delta}\right) e^{-(1-\alpha)(n+g+\delta)t} $

  • And we can then piggyback on the great mathematicians of the past…

 

Remember Where We Came From…

  • This is the complete specification of the Solow model base case:
    • A production function where a 1% increase in K/L brings an α% increase in Y/L
    • A production function that also includes the efficiency of labor E
    • Labor force growth at constant proportional rate n
    • Efficiency of labor growth at constant proportional rate g
    • Capital accumulation equation: savings and depreciation

(5)    $ \left(\frac{Y}{L}\right) = \left(\frac{K}{L}\right)^{\alpha}(E)^{1-\alpha} $

(6)    $ \frac{1}{L}\frac{dL}{dt} = n $

(7)    $ \frac{1}{E}\frac{dE}{dt} = g $

(8)    $ \frac{1}{K}\frac{dLK}{dt} = s\left(\frac{Y}{K}\right) - \delta $


 

Catch Our Breath…

delong

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