RCK Model

Why study this model?

Model ingredients: households, firms, governments

Households:

Firms:

Solving the model: 

Need a budget constraint before maximization problem. Will be lifetime consumption cannot exceed the sum of initial wealth plus the present discounted value of lifetime labor. No-Ponzi-Game.

The utility function: \(max[A(0)^{1 - \theta} \frac{L(0)}{H} \int_{t = 0} ^ \infty e^{-\rho t} e^{(1 - \theta)gt} e^{nt} \frac{c(t)^{1 - \theta}}{1 - \theta}dt]\)

Budget constraint: presented discounted value of utility function. 
Optimal consumption growth (KNOW THIS EQUATION): \(\frac{\dot C}{C} = \frac{r(t) - \rho}{\theta}\)
Use log utility function over two periods:
\(U(C_1, C_2) = log(C_1) + (\frac{1}{1 + \rho}) log(C_2)\)
Present discounted value of consumption = present discounted value of output:
\(C_1 + (\frac{1}{1 + r})C_2 = Y_1 + (\frac{1}{1 + r}) Y_2\)
For this problem, use the Lagrangian form. The first order conditions are:
\(\frac{1}{C_1} = \lambda\)
\(\frac{1}{1 + \rho} \frac{1}{C_2} = \frac{\lambda}{1 + r}\)
\(C_1 + (\frac{1}{1 + r})C_2 = Y_1 + (\frac{1}{1 + r}) Y_2\)
The results\(\frac{C_2}{C_1} = (\frac{1 + r}{1 + \rho})\)
*NB: \(\log(1 + tiny) \approx tiny\) and \(C_2 = C_1 + \Delta C_1\)
How? Subsitute: \(\frac{C_1 + \Delta C_1}{C_1} = \frac{1 + r}{1 + \rho}\)
\(1 + \frac{\Delta C_1}{C_1} = \frac{1 + r}{1 + \rho}\)
Take logs: \(\log(1 + \frac{\Delta C_1}{C_1}) = \log(1 + r) + \log(1 + \rho)\)
Using the "tiny log" rule: \(\frac{\Delta C}{C_1} = r - \rho\)
Now, when we look at consumption per effective worker:
\(\frac{\dot c}{c} = \frac{f'(k(t)) - \rho - \theta g}{\theta}\)
\(\dot k(t) = f(k(t)) - c(t) - (n + g) k(t)\)
Should look familiar because just like the Solow model, but with savings endogenized as income minus consumption minus the growth of A and L but no depreciation. At the end of the day, we've just brought in a second differential equation for consumption since we use it to explain savings. 
\(\therefore\) We have a system of differential equations that cannot be solved analytically. Thus we must observe with graphs. 
SEE GRAPHS  of dynamics of c and k