CLASS #18: THE SOLOW GROWTH MODEL

• Modeling growth: the Solow growth model.
• Convergence toward the steady state in the Solow growth model ... or ``the last will be first''.
• The Solow growth model with population growth.

Mechanics of the Solow growth model.

Assumptions:

• Closed economy:
  
  $$Y_t=C_t+I_t \mbox{ }\rightarrow \mbox{ } S_t=I_t$$
  
  

• Cobb Douglas production function
• Saving (hence Investment) is a constant fraction s of output:
  I_t=sY_t
• Total factor productivity (A) is constant.
• For the time being, population is constant:
  
  N_t=N
  
  
  .

From the definition of investment:

$$K_{t+1}=(1-\delta)K_t+I_t$$

$$=(1-\delta)K_t+sY_t$$

Divide both sides by N and obtain:

$$k_{t+1}=(1-\delta)k_t+sAk_t^\alpha$$

or

k_t+1=f(k_t).

This is a non-linear difference equation: strange object! Let us study the function $f(k)=(1-\delta)k+sAk^\alpha$:
1) f(0)=0

2) $\lim_{k\rightarrow\infty} f(k)=\infty$

3) $f'(k)=(1-\delta)+sA\alpha k^{\alpha-1}>0$,

4) $\lim_{k\rightarrow \infty} f'(k)=(1-\delta)<1$,
$\lim_{k\rightarrow 0} f'(k)=\infty>1$

5) $f''(k)=sA\alpha(\alpha-1)k^{\alpha-2}<0$

• Steady state per capita level of capital k^*.
• Convergence to k^* from any level of capital.
• A steady state per capita level of capital implies a steady state level of output $y^*=Ak^{*\alpha}$: the economy eventually stops growing.
• What does the steady state depend upon?
  
  $$k^*=(\frac{sA}{\delta})^{\frac{1}{1-\alpha}}$$
  
  

• How fast do you reach the steady state?

Consider population growth at a rate $\lambda$:

$$N_{t+1}=(1+\lambda)N_t$$

divide both sides of

$$K_{t+1}=(1-\delta)K_t+sY_t$$

by N_t and obtain:

$$(1+\lambda)k_{t+1}=(1-\delta)k_t+sAk_t^\alpha$$

or

$$k_{t+1}=\frac{1-\delta}{1+\lambda}k_t+\frac{sA}{1+\lambda}k_t^\alpha$$

• The steady state per capita level of capital is lower:
  
  • because of population growth, you have to provide capital for the newborn, and that capital depreciates.
• Again, convergence to k^* from any level of capital; and again, a steady state per capita level of capital implies a steady state level of output $y^*=Ak^{*\alpha}$: the economy eventually stops growing.
• The new steady state is:
  
  $$k^*=(\frac{sA}{\delta+\lambda})^{\frac{1}{1-\alpha}}$$
  
  
  and depends:
  • positively on the saving rate s, and A;
  • negatively on depreciation and population growth (remember the empirical regularities about growth);

This model does not explain sustained growth - in the next class we will introduce technological progress.