$$ \tilde g_{Y/L} - g_{Y/L} = \frac{\beta b + (\beta+\gamma)n}{1-\alpha} $$ Quantifying the drag (cont'd) ======================================================== Calibration according to Nordhaus (1992): ```{r} beta = 0.1; gamma = 0.1; alpha = 0.2; n = 0.01; b = 0.005 drag = (beta * b + (beta + gamma) * n) / (1 - alpha) drag ``` The limited land and decreasing nonrenewable resources reduce per-capita growth rate by **0.3 percentage point**. A more elaborate model of Nordhaus's estimates smaller drag. The drag is not trivial but not large. Supplement: Stability ========================================================= We analyzed the BGP for the baseline Solow model because it has **global stability**. Start from anywhere, you eventually reach that path (in very long run). As for the environmental Solow model, we bypassed the analysis of transition path and got directly to the BGP. We still don't know whether or not this BGP will is reached in the long run. Some macroeconomic analyses are based on unstable equilibrium paths .... but that is a bad practice. I would strongly recommend to check **stability** somehow. We do this by a computer simulation. It's not rigorous mathematical proof but we can be sure that the above BGP analysis is relevant. Simulation ========================================================= ```{r, message=FALSE} library(dplyr) s = 0.3; alpha = 0.2; beta = 0.1 gamma = 0.1; delta = 0.05; b = 0.005 g = 0.02; n = 0.01; dt = 0.05; t = seq(0.0, 250.0, by=dt) df = data.frame(t) df = df %>% mutate(A = exp(g * t)) %>% mutate(L = exp(n * t)) %>% mutate(T = exp(0 * t)) %>% mutate(E = b * exp(-b * df$t)) head(df) ``` Simulation (cont'd) ========================================================= ```{r} F = function(K, A, L, T, E){ Y = (K^alpha) * (E^beta) * (T^gamma) * (A*L)^(1-alpha-beta-gamma) return(Y) } K = 0.05 for (i in 1:nrow(df)){ A = df[i, "A"] L = df[i, "L"] T = df[i, "T"] E = df[i, "E"] Y = F(K, A, L, T, E) df[i, "K"] = K df[i, "Y"] = Y K = K + dt * (s * Y - delta * K) } head(df) ``` Simulation (cont'd) ========================================================= ```{r, fig.width=12, fig.asp=0.75, fig.align='center'} library(ggplot2) ggplot(data=df, aes(x=t, y=Y)) + geom_line() + scale_y_log10() + theme(text=element_text(size=30)) ``` **It seems like the equilibrium path is converging to the BGP.** Simulation (cont'd) ========================================================= :small-code ```{r, fig.width=12, fig.asp=0.75, fig.align='center'} growth.rate = (df[2:nrow(df), "Y"] - df[1:(nrow(df)-1), "Y"]) / df[1:(nrow(df)-1), "Y"] / dt qplot(x=df[1:(nrow(df)-1),"t"], y=growth.rate, geom='line') + geom_hline(aes(yintercept=((1-alpha-beta-gamma) * (g + n) - beta * b )/ (1 - alpha)), color="red") + labs(y='Growth rate of Y', x='time') + theme(text=element_text(size=30)) ``` Ramsey-Cass-Koopmans Model ========================================================= In the Solow model, the saving function is constant. In the Ramsey-Cass-Koopmans model (or more simply Ramsey model), - the amount of saving is endogenously determined - for a certain specification, we can derive the constant saving rate; the Solow model with **micro-foundation**. The Ramsey model predicts - that the growth rate for per-capita income/capital is determined solely by exogenous technology growth rate. - that is, the **prediction of the Solow model is robust**. Ramsey-Cass-Koopmans Model (cont'd) ========================================================= There is a subtle difference concerning policy change. - In the Solow model, a rise of government purchases crowds out investment. - In the Ramsey model, there is not crowding-out effect. This difference comes from the fact that the agents in the Ramsey model behave forward-lookingly and take there future income as given. A permanent policy change alters the total income but doesn't raise substitution between saving and consumption. Simple Optimal Growth Model ========================================================= Before tackling the model in Romer 4e, we analyze the **one-sector optimal growth model**: $$ \begin{aligned} \max \int_0^\infty e^{-\rho t} u(c(t)) dt \end{aligned} $$ subject to $$ \begin{aligned} \dot k(t) &= f(k(t)) - \delta k(t) - c(t)\\ k(t) &\ge 0\\ c(t) &\ge 0 \end{aligned} $$ - $\rho$: discount rate - $u$: utility function **Maximize discounted sum of utility from consumption stream subject to the capital accumulation constraint** Simple Optimal Growth Model (cont'd) ========================================================= We will study - how to derive this continuous-time optimization from much easier-to-understand discrete-time analog, - how to solve it analytically (Hamiltonian) - how to relate it to the model in Romer 4e - how to solve Ramsey model numerically (Dynamic Programming)