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