RCK Model
Why study this model?
- Solow was too restrictive with exogenous savings
- Most important Exogenous growth based on micro-foundations
- Basis of modern Dynamic (Stochastic) General Equilibrium models (DSGE)
- Permits the occurrence of business cycles
- Because of micro foundations, we can do welfare analysis
Model ingredients: households, firms, governments
Households:
- Large number of identical
- Grows at rate n
- Owner of labor returns
- Owner of capital returns
- Maximize infinitely-lived utility function
Firms:
- At every moment in time they use capital and labor to make output
- Constant returns to scale-- factors of production are paid their marginal product
- Economy is competitive so zero profits
- Real interest rate is equal to the marginal product of capital (K)
- Wage per unit of effective labor is equal to the marginal product of labor (AL)
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: \(max[A(0)^{1 - \theta} \frac{L(0)}{H} \int_{t = 0} ^ \infty e^{-\rho t} e^{(1 - \theta)gt} e^{nt} \frac{c(t)^{1 - \theta}}{1 - \theta}dt]\)
- Constant, relative risk aversion is \(1 - \theta\) (not derived from consumption)
- \(e^{- \rho t}\): assumes present consumption is more desirable, so discount future utilities. Rate at which you discount that future utility is called the discount rate. i.e. bringing future dollars into today's dollars with present value
- \(e^{(1 - \theta)gt}\): assuming exogenous growth rate of technology (A) with g
- \(e^{nt}\): assuming exogenous growth rate of population (L) with n
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