class: center, middle, inverse, title-slide # Ramsey Model ## Macroeconomics ### Kenji Sato ### Day 09 --- class: center, middle # Departure from the Solow model --- ## Question Suppose that you have just heard that your tax you had already paid (say, 10,000 Yen) will be reimbursed next month. What would you do with the expected windfall gains? 1. Nothing. Wait until you have the cash. 2. Go shopping or go for a drink this weekend, expecting the refund. --- ## Why Solow not enough? In the Solow model, **the saving rate is exogenous**. Take a tax reduction for example. In the Solow model, consumers raises consumption at the time of enforcement of the new tax system. But usually governments would inform the change of tax well before the enforcement and we might drink a toast to the tax reduction. Similarly, in case of tax hike, it is possible that consumers consume less immediately after knowing that they will have to pay more to the government. --- ## Why Solow not enough? (Cont'd) So, the Solow model is missing an important aspect of our economic activity: **Expectation formation**. Although rigorously dealing with the problem of expectations is beyond the scope of this course, the **Ramsey model** will serve as the skeleton of more elaborate models. --- ## Is saving rate constant independent of income? Evidence suggests that the saving rate on current income is roughly increasing in income. (e.g., [Dynan, Skinner and Zeldes 2004](http://www.journals.uchicago.edu/doi/abs/10.1086/381475)) The more you earn, the lower the fraction of spending gets (the higher the fraction of saving gets). The Solow assumption, `$$S = sY$$` with constant `\(s\)`, might be critical to the predictions by the Solow model. --- ## Note In the following discussion, we will follow the following rule to choose symbols. For example, * Upper-case = Aggregate variables - `\(C\)`: aggregate consumption - `\(K\)`: total capital * Lower-case = Per-capita variables - `\(c\)`: consumption per capita - `\(k\)`: capital per capita * Lower-case with hat = Per unit of effective labor - `\(\hat{\!c}\)`: consumption per unit of effective labor - `\(\hat{\!k}\)`: capital per unit of effective labor --- class: center, middle # Ramsey model --- ## Endogenize consumption-saving behavior Each individual decides when and how much to consume by solving the utility maximization: `$$\max \int_0^\infty e^{-\rho t} u(c(t))dt$$` subject to `$$c(t) + \dot{s}(t) = r(t) s(t) + w(t)$$` For the moment let's suppose no population growth (`\(n = 0\)`) --- ## Instantaneous utility: `\(u\)` At time `\(t\)`, a representative consumer consumes `$$c(t).$$` From the consumption, they get utility at that time `$$u(c(t))$$` `\(u\)` is increasing with `\(c\)` and has diminishing marginal utility. `$$u' > 0 \quad \text{and}\quad u'' < 0$$` --- ## Discount If you have a choice, which would you choose? * $100 given now * $100 given in one month Or, you might be indifferent. --- ## Discount (cont'd) Future consumption, anticipated now, is discounted because you must wait. The same amount of consumption, `\(\bar c\)`, enjoyed at different time points, gives higher _present_ utility (measured at `\(t = 0\)`), when the consumption takes place in the nearer future. --- ## Exponential discounting `\(\rho > 0\)` We assume that the consumers have discounted instantaneous utility of the following form: `$$e^{- \rho t} u(c),$$` where `\(\rho > 0\)` is called the discount rate. How strongly the consumers discount the future utility is controlled by a new parameter `\(\rho > 0\)`. --- ## Quiz `$$e^{-\rho t} u(c)$$` Which is a correct statement? * Consumers with High `\(\rho\)` are more patient than those with Low `\(\rho\)`. * Consumers with Low `\(\rho\)` are more patient than those with High `\(\rho\)`. --- ## Total utility Given the representative consumer's consumption stream, `$$t \mapsto c(t),$$` they expect the total utility `$$\int_0^\infty e^{-\rho t} u(c(t))dt$$` From two consumption streams, `\(c'\)` and `\(c''\)`, they choose `\(c'\)` if `$$\int_0^\infty e^{-\rho t} u(c'(t))dt \ge \int_0^\infty e^{-\rho t} u(c''(t))dt$$` --- ## Maximization problem The consumer chooses the best consumption stream that miximize the total utility: `$$\max \int_0^\infty e^{-\rho t} u(c(t))dt$$` The consumer doesn't have the luxury of choosing their consumption stream completely freely. --- ## Budget constraint #### Cash inflow * He/she supplies one unit of labor and earns `\(w(t)\)`. * Each indiviual earns interest income, `\(r(t) s(t)\)`, from his/her savings, `\(s(t)\)`. `\(r(t)\)` is the interest rate. #### Cash outflow * Consumption `\(c(t)\)`. * The amount of saving/dissaving at time `\(t\)` is `\(\dot{s}(t)\)`. `\(\dot{s}(t) > 0\)` when depositing, `\(\dot{s}(t) < 0\)` when withdrawal. --- ## Budget constraint At any point in time, `\(c\)` must satisfy the following **flow budget constraint**: `$$c(t) + \dot{s}(t) = r(t) s(t) + w(t),$$` subject to which the consumer solves the problem `$$\max \int_0^\infty e^{-\rho t} u(c(t))dt$$` --- ## No-Ponzi Game condition `$$c(t) + \dot{s}(t) = r(t) s(t) + w(t)$$` Since `\(\dot{s}\)` can be negative, they can borrow money to consume more than they earn. What if they could borrow as much money they you wanted and had no obligation to pay back? They would increase consumption to get more utility from consumption, resulting in infinitely large utility. We must assume some condition to avoid this possibility. --- ## Integrated flow budget constraint From the flow budget constraint, we have `$$e^{- R(t)} c(t) + e^{-R(t)} \dot{s}(t) = e^{-R(t)} r(t) s(t) + e^{-R(t)} w(t),$$` where `$$R(t) = \int_0^t r(s)ds \Longrightarrow \dot{\!R}(t) = r(t)$$` Integrate both side to get `$$\begin{multline} \int_0^\infty e^{-R(t)} c(t) dt + \int_0^\infty e^{-R(t)} \dot{s}(t) dt\\ = \int_0^\infty e^{-R(t)} r(t) s(t) dt + \int_0^\infty e^{-R(t)} w(t) dt, \end{multline}$$` --- ## Integrated flow budget constraint (cont'd) Notice that `$$\begin{aligned} \int_0^\infty e^{- R(t)} \dot{s}(t)dt = \lim_{t \to \infty} e^{- R(t)} s(t) - s(0) + \int_0^\infty r(t) e^{- R(t)} s(t) dt \end{aligned}$$` by integration by parts formula. We get `$$\int_0^\infty e^{- R(t)} c(t) dt + \lim_{t \to \infty} e^{- R(t)} s(t) = s(0) + \int_0^\infty e^{-R(t)} w(t) dt$$` --- ## No-Ponzi Game condition, mathematically stated `$$\int_0^\infty e^{- R(t)} c(t) dt + \lim_{t \to \infty} e^{- R(t)} s(t) = s(0) + \int_0^\infty e^{-R(t)} w(t) dt$$` If it holded that `$$\lim_{t \to \infty} e^{- R(t)} s(t) < 0$$` the consumers can consume more than they earn over their life time. It might be natural to assume `$$\lim_{t \to \infty} e^{- R(t)} s(t) \ge 0$$` --- ## Life-time budget constraint `$$\int_0^\infty e^{- R(t)} c(t) dt = s(0) + \int_0^\infty e^{-R(t)} w(t) dt$$` This constraint is called **life-time budget constraint**. * The right hand-side is initial asset plus discounted sum of labor income; * left hand-side the discounted sum of life-time consumption. --- ## Consumption maximization restated We can restate the consumer's problem to `$$\max \int_0^\infty e^{-\rho t} u(c(t))dt$$` subject to `$$\int_0^\infty e^{- R(t)} c(t) dt = s(0) + \int_0^\infty e^{-R(t)} w(t) dt$$` Remark: In Romer's textbook, a solution method using a Lagrangian for this optimization problem is discussed. We will instead study the Hamiltonian method applied to the original problem. The latter technique will turn out to be useful when you read research papers. --- ## Dynamics of the model We will analyze the development of two endogenous variables `\(\hat{k}\)` and `\(\hat{c}\)` governed by the two-dimensional differential equation. `$$\begin{aligned} \dot{\hat{k}} &= f\left(\hat{k}\right) - \hat{c} - (\delta + g + n)\hat{k}\\ \dot{\hat{c}} &= \text{(see it later)} \end{aligned}$$` Let's do the problem for today as an exercise.