class: center, middle, inverse, title-slide # Ramsey Model: Complete Description ## Macroeconomics (2017Q2) ### Kenji Sato ### 2017-07-10 --- <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { Macros: { } } }); </script> $$\let\oldhat\hat \renewcommand{\hat}[1]{\oldhat{\hspace{0pt} #1}}$$ <div style="margin-top: -2em"></div> ## Homework / Exercises Visit [kjst.jp/ma17q2pr](http://kjst.jp/ma17q2pr) now to check your submission status. Make sure all your submissions got purple icons. ### Midterm Report Get invitation for `mid`. Visit [kjst.jp/ma17q2hw](kjst.jp/ma17q2hw) > Due 2017-07-14 18:00. > Hand in by Pull Request. --- ## Exogenous variables Population growth `$$\dot{L} = n L$$` Technical growth `$$\dot{A} = g A$$` --- class: center, middle # consumers --- ## Consumer Behavior: Objective Consumers act altruistically. They care not only about their own consumption but also about offspring's. Each individual maximizes the total utility: `$$\max \int_0^\infty e^{-\rho t} u(c(t)) e^{nt} dt$$` Since there are `\(L(0)\)` consumers at `\(t = 0\)`, the economy as a whole miximizes the following objective. `$$\max \int_0^\infty e^{-\rho t} u(c(t)) L(t) dt$$` --- ## Consumer Behavior: Budget constraint Each consumer is subject to the following flow budget constraint: `$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t) + w(t)$$` We have `\(-n s(t)\)` in the interest term because of the population growth. Per-person share of the interest income is reduced by the factor of growth rate. --- ## Consumer Behavior: First-order condition The representative consumer solves the following maximization problem: `$$\max \int_0^\infty e^{-(\rho - n) t} u(c(t)) dt$$` subject to `$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t) + w(t)$$` **Problem: Set up the Hamiltonian. (3min)** --- ## Consumer Behavior: Canonical Equations `$$\mathcal H = e^{-(\rho - n) t} u(c) + \lambda [(r - n) s + w - c]$$` **Problem: Compute the canonical equations. (5min)** 1. `\(\partial \mathcal H / \partial c = 0\)` 2. `\(\dot{\lambda} = - \partial \mathcal H / \partial s\)` 3. `\(\dot{s} = \partial \mathcal H / \partial \lambda\)` --- ## Consumer Behavior: Euler equation **Problem: Derive the Euler equation. (5min)** `$$\begin{aligned} \frac{\dot{c}(t)}{c(t)} = \frac{r(t) - \rho}{\Theta(c)} \end{aligned}$$` Under the assumption of CRRA utility `$$u(c) = \frac{c^{1 - \theta} - 1}{1 - \theta},\quad \theta > 0$$` the Euler equation is expressed as `$$\begin{aligned} \frac{\dot{c}(t)}{c(t)} = \frac{r(t) - \rho}{\theta} \end{aligned}$$` --- ## Euler equation for `\(C/(AL)\)` Define the consumption per unit of effective labor: `$$\hat{c} = \frac{c}{A} = \frac{C}{AL}$$` Since `\(A\)` grows exogenously with a constant growth rate `\(g\)`, `$$\frac{\dot{\hat{c}}}{\hat{c}} = \frac{\dot{c}(t)}{c(t)} - g$$` We therefore have `$$\frac{\dot{\hat{c}}}{\hat{c}} = \frac{r(t) - \rho - \theta g}{\theta}$$` --- class: center, middle # firms --- ## Firm behavior: Production function Let's describe the behavior of the firms. This is mostly equivalent to the setup of the Solow model. The firms have access to the CRS production technology `$$Y(t) = F(K(t), A(t) L(t)).$$` In per-`\(AL\)` term, `$$\hat{y}(t) = f\left(\hat{k} (t)\right) = \frac{F(K, AL)}{AL}$$` --- ## Firm behavior: Profit maximization Rent `$$r(t) + \delta = f'\left( \hat{k}(t) \right)$$` Wage `$$w(t) = A(t) \left[ f\left( \hat{k}(t) \right) - k(t) f'\left( \hat{k}(t) \right) \right]$$` Wage per effective labor `$$\hat{w}(t) = f\left( \hat{k}(t) \right) - k(t) f'\left( \hat{k}(t) \right)$$` --- ## Equilibrium: Capital Accumulation The capital market clearing condition: `$$s(t) L(t) = K(t)$$` The flow budget constraint `$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t) + w(t)$$` is transformed into `$$\frac{\dot{\hat{k}}}{\hat{k}} = \frac{\dot s}{s} - g = \frac{f\left(\hat{k}\right) - \hat{c} - (\delta + g + n) \hat{k}}{\hat{k}}$$` --- ## Equilibrium: First-order Dynamics We have the following capital accumulation equation: `$$\dot{\hat{k}} = f\left(\hat{k}\right) - \hat{c} - (\delta + g + n) \hat{k}$$` The dynamics of the Ramsey model is determined by this capital accumulation equation together with the Euler equation: `$$\frac{\dot{\hat{c}}}{\hat{c}} = \frac{f'\left(\hat{k}\right) - \delta - \rho - \theta g}{\theta}$$`