Properties
- These dynamics imply that the economy will get to a steady state
- Only the stable arm will get you to the path of steady state -- saddle-pack
Looking at the Steady State
- Once the economy reaches an equilibrium, the steady state properties become like Solow
- Output, capital, and consumption per effective worker grow at the same rate of zero
- Savings is also constant-- just like assumption of Solow
Golden Rule level of k:
- Because of the way the model is set up, it is not possible to attain the golden rule level of k -- will necessarily be to the left of k*
- Remember that the golden rule implied that the level of consumption is maximized at the level of k
- Technically, cannot reach because of the e exponents in the utility function:\(e^{(- \rho + (1 - \theta)g + n)t}\) we know that \(- \rho + (1 - \theta)g + n < 0 \Longrightarrow n + g < \rho + \theta g\). \(f'(k_{GR}) = n + g; f'(k*) = \rho + \theta g\) THUS we are unable to meet the Golden Rule (this is very telling... Someone is funny)
A change in the discount rate:
- A fall in \(\rho\) (discount rate) is equivalent to increase in savings
- What are the qualitative effects of such a shock?
- Only one of the differential equations is affected:
\(\frac{\dot c}{c} = \frac{f'(k) - \rho - \theta g}{\theta}\)
What happens to the graph? You need f'(k) to fall to keep consumption change equal to zero so the line moves to the right as k increases (decreasing slope)
SEE GRAPH IN NOTES
*NB: When moving in the short-run, the only variable that changes quickly is consumption (c) so this will absorb most of the shock (k takes time to build up)
Introducing the government
- Government spending is about 20-25% of GDP
- Important to consider ho the model changes with the introduction of G, the government sector
- Higher value of G(t) shifts the \(\dot k = 0\) line down
- Government consumption crowds out private consumption
- What happens next depends on whether G is temporary or permanent
SEE GRAPH IN NOTES